The maximum size of a Sidon set contained in a sparse random set of integers Yoshiharu Kohayakawa

†

Sangjune Lee

Abstract A set A of non-negative integers is called a Sidon set if all the sums a1 + a2 , with a1 ≤ a2 and a1 , a2 ∈ A, are distinct. One of the best studied problems on Sidon sets is the determination of the maximum possible size F (n) of a Sidon subset of [n] = {0, 1, . . . , n − 1}. Thanks to results of Chowla, Erd˝ os and Tur´a√ n from the 1940s, it is known that F (n) = (1 + o(1)) n. In this paper we study Sidon subsets of sparse random sets of integers, replacing the ‘dense environment’ [n] by a sparse, random subset R of [n], and ask how large a subset S ⊂ R can be, if we require that S should be a Sidon set. Let R = [n]m be a random
subset of [n] of cardinaln ity m = m(n), with all the m subsets of [n] equiprobable. We investigate the random variable F ([n]m ) = max |S|, where the maximum is taken over all Sidon subsets S ⊂ [n]m , and obtain quite precise information on F ([n]m ) for the whole range of m. An abridged version of our results states as follows. Let 0 ≤ a ≤ 1 be a fixed constant and suppose m = m(n) = (1 + o(1))na . We show that there is a constant b = b(a) such that, almost surely, we have F ([n]m ) = nb+o(1) . As it turns out, the function b = b(a) is a continuous, piecewise linear function of a that is non-differentiable at two points: a = 1/3 and a = 2/3. Somewhat surprisingly, between those two points, the function b = b(a) is constant. 1 Introduction Recent years have witnessed vigorous research in the classical area of additive combinatorics. An attractive feature of these developments is that applications in theoretical computer science have motivated some of the striking research in the area (see, e.g., [26]). For a modern treatment of the subject, the reader is referred to [25]. In this paper, we investigate a natural problem ∗ The first author was partially supported by CNPq (Proc. 308509/2007-2). The third author was supported by the NSF grant DMS 0800070. † Instituto de Matem´ atica e Estat´ıstica, Universidade de S˜ ao Paulo, Brazil; [email protected] ‡ Department of Mathematics and Computer Science, Emory University, Atlanta, USA; [email protected], [email protected]

‡

Vojtˇech R¨odl

∗

‡

in probabilistic additive combinatorics. Among the best known concepts in additive number theory is the notion of a Sidon set. We investigate Sidon subsets of random sets of integers, and obtain surprisingly tight bounds on their relative density. A set A of non-negative integers is called a Sidon set if all the sums a1 + a2 , with a1 ≤ a2 and a1 , a2 ∈ A, are distinct. One of the best studied problems on Sidon sets is the determination of the maximum possible size F (n) of a Sidon subset of [n] = {0, 1, . . . , n − 1}. In 1941, √ Erd˝os and Tur´an [7] showed that F (n) ≤ (1 + o(1)) n. In 1944, Chowla [3] and Erd˝os [6], independently of each √ other, proved that F (n) ≥ (1 + o(1)) n. Consequently, √ it is known that F (n) = (1 + o(1)) n. For a wealth of related material, the reader is referred to the classical monograph of Harberstam and Roth [10] and to a recent survey by O’Bryant [19] and the references therein. Here, we investigate Sidon subsets of sparse, random sets of integers, that is, we replace the ‘environment’ [n] by a sparse, random subset R of [n], and ask how large a subset S ⊂ R can be, if we require that S should be a Sidon set. Investigating how classical extremal results in ‘dense’ environments transfer to ‘sparse’ settings has proved to be a deep line of research. A fascinating example along these lines occurs in the celebrated work of Tao and Green [9], where Szemer´edi’s classical theorem on arithmetic progressions [24] is transferred to certain sparse, pseudorandom sets of integers (see [20, 21, 25] for more in this direction). Much closer examples to our setting are a version of Roth’s theorem on 3-term arithmetic progressions [22] for random subsets of integers [17], and the far reaching generalizations due to Schacht [23] and Conlon and Gowers [4]. For the sake of brevity, we shall not discuss this further and refer the reader to [23], [11, Chapter 8], and [15, Section 4]. Let us now state a weak, but less technical version of our main result. Let F (R) = max |S|, where the maximum is taken over all Sidon subsets S ⊂ R. Let [n]m be a random
subset of [n] of cardinality m = n m(n), with all the m subsets of [n] equiprobable. We are interested in the random variable F ([n]m ). Standard methods give that, almost surely, that is, with probability tending to 1 as n → ∞, we

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b

factor for the range of m ≤ n2/3−δ for any fixed δ > 0; for the remaining values of m, we shall not be off by more than a polylogarithmic factor. On the other hand, understanding the fine behavior of F ([n]m ) remains open. For example, it would be interesting to decide whether, for any constant α, there is a constant β = β(α) such that, if m = αn1/3 , then F ([n]m ) = (β + o(1))m holds almost surely. Also it would be interesting to determine the behavior of F ([n]m ) for m ≥ n2/3−δ up to a constant multiplicative factor.

F ([n]m ) = nb+o(1) for m = na 1/2 1/3

1/3

2/3

1

a

2 Main results 2.1 Statement of the main results. Our first result corresponds to the range 0 ≤ a ≤ 1/3 in Theorem 1.1.

Figure 1: The graph of b = b(a)

have F ([n]m ) = (1 − o(1))m if m = m(n)  n1/3 . On Theorem 2.1. For m = m(n)  n1/3 , we almost the other hand, the results of Schacht [23] and Conlon surely have and Gowers [4] imply that, if m = m(n)  n1/3 , then, almost surely, we have (2.4) F ([n]m ) = m(1 − o(1)). (1.1)

Our next result covers the range 1/3 ≤ a < 2/3 in Theorem 1.1.

F ([n]m ) = o(m).

Thus F ([n]m ) undergoes a sudden change of behaviour at m = n1/3+o(1) . The following abridged version of Theorem 2.2. For any 0 < δ < 1/3, there is a positive 1/3 ≤ m = m(n) ≤ our results already gives us quite precise information constant c2 = c2 (δ) such that if 2n 2/3−δ n , then, almost surely, we have on F ([n]m ) for the whole range of m. (2.5)
1/3
1/3 Theorem 1.1. Let 0 ≤ a ≤ 1 be a fixed constant. c1 n log(m3 /n) ≤ F ([n]m ) ≤ c2 n log(m3 /n) , Suppose m = m(n) = (1 + o(1))na . There exists a where c1 is a positive absolute constant. constant b = b(a) such that almost surely We now turn to the point a = 2/3 in Theorem 1.1.

F ([n]m ) = nb+o(1) .

(1.2)

Theorem 2.3. For any 0 ≤ δ < 1/3, there is a positive constant c3 = c3 (δ) such that if 1 ≤ α = α(n) ≤ nδ and

Furthermore,

(1.3)

  a b(a) = 1/3   a/2

if 0 ≤ a ≤ 1/3, if 1/3 ≤ a ≤ 2/3, if 2/3 ≤ a ≤ 1.

(2.6)

Thus, the function b = b(a) is piecewise linear. The graph of b = b(a) is given in Figure 1. The point (a, b) = (1, 1/2) in the graph is clear from the Erd˝ os–Tur´an and Chowla results [3, 6, 7] mentioned above. The behaviour of b = b(a) in the interval 0 ≤ a ≤ 1/3 is not hard to establish. The fact that the point (1/3, 1/3) could be an interesting point in the graph is suggested by the results of Schacht [23] and Conlon and Gowers [4]. It is somewhat surprising that, besides the point a = 1/3, there is a second value at which b = b(a) is “critical,” namely, a = 2/3. Finally, we find it rather interesting that b = b(a) should be constant between those two critical points. In Section 2, we state our results in full. We in fact determine F ([n]m ) up to a constant multiplicative

m = m(n) =

1 2/3 n (log n)2/3 , α

then, almost surely, we have (2.7) c3 (n log n)1/3 ≤ F ([n]m ) ≤ c4 (n log n)1/3

log n , log(1 + α)

where c4 is a positive absolute constant. We remark that Theorems 2.2 and 2.3 consider ranges that overlap (functions m = m(n) of the 0 form n2/3−δ for some 0 < δ 0 < 1/3 are covered by both theorems). Finally, we consider the range 2/3 ≤ a ≤ 1 in Theorem 1.1. Theorem 2.4. There exist positive absolute constants c5 and c6 for which the following holds. If n2/3 (log n)2/3 ≤ m = m(n) ≤ n2/3 (log n)5/3 , then √ (2.8) c5 m ≤ F ([n]m ) ≤ c6 n1/3 (log n)4/3

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almost surely. Furthermore, if n2/3 (log n)5/3 ≤ m = over all mod-n-Sidon sets S ⊂ R. We shall simply m(n) ≤ n, then, almost surely, write F mod (R) for Fnmod (R) if there is no danger of confusion. p √ (2.9) c5 m ≤ F ([n]m ) ≤ c6 m log n. The following lemma may be used to translate results from the modular setting to the non-modular 2.2 Organization. The next section is devoted to setting. some preliminaries. In Section 4, we consider the upper bounds in Theorems 2.2–2.4. Section 5 is devoted Lemma 3.2. For any R ⊂ [n], we have to the proof of the key lemma that is required in the proofs of those upper bounds. Section 6 contains (3.10) F mod (R) ≤ F (R) ≤ 2F mod (R). a key lemma for the proofs of the lower bounds in Theorems 2.2–2.4. The proof of Theorem 2.1, which Taking R = [n]p , Lemma 3.2 easily implies the is based on a deletion argument, is given in Section 7. following corollary. In Section 8, we consider the proofs of the lower bounds in Theorems 2.2–2.4. Corollary 3.1. Suppose a(n), b(n) and 0 ≤ p = For simplicity, we omit “floor” and “ceiling” sym- p(n) ≤ 1 are functions of n for which a(n) ≤ bols in our formulae when they are not essential. F mod ([n]p ) ≤ b(n) holds with overwhelming probability. Then a(n) ≤ F ([n]p ) ≤ 2b(n) also holds with over3 Preliminaries whelming probability. 3.1 The uniform model and the binomial model. Instead of working with the uniform 3.3 Monotonicy. In this section, we give two monomodel [n]m of random subsets of [n], it will be tonicity results. First, one can easily observe the followmore convenient to work with the so called bino- ing fact (see, e.g., [12, Lemma 1.10]). mial model [n]p , which has more “independence”. Fix 0 < p = p(n) ≤ 1. Each element of [n] is put Fact 3.1. Let p = p(n) and q = q(n) be such that 0 ≤ in [n]p with probability p, independently of all other p < q ≤ 1, and let a(n) > 0 and b(n) > 0 be functions elements. As usual, results on either model may often of n. be translated to results in the other model (for instance, mod ([n]p ) ≥ a(n) holds with overwhelming see [2, Section 2.1] and [12, Section 1.4]). Before we (i) If F probability, then F mod ([n]q ) ≥ a(n) holds with proceed, let us make the following definition. overwhelming probability. Definition 3.1. We shall say that an event in the probability space of the random sets [n]p or in the (ii) If F mod ([n]q ) ≤ b(n) holds with overwhelming probability space of the random sets [n]m holds with probability, then F mod ([n]p ) ≤ b(n) holds with overwhelming probability if the probability of failure of overwhelming probability. that event is O(n−C ) for any constant C, that is, if the probability of failure is superpolynomially small. Statements (i) and (ii) in Fact 3.1 are, in fact, For us, the following consequence of Pittel’s inequality equivalent. We state them both explicitly just for will suffice for translating results on [n]p to results on convenience. Our second monotonicity result is, in some sense, a converse to Fact 3.1. [n]m . Lemma 3.1. Let 1 ≤ m = m(n) < n and p = p(n) be such that np = m. Let P be an event in the probability space of the random sets [n]p . If [n]p has P
with overwhelming probability, then [n]m has P ∩ [n] m with overwhelming probability.

Lemma 3.3. Let p = p(n) and q = q(n) be such that 0 ≤ p < q ≤ 1. Let a(n) > 0 be a function of n such that pna(n)  log n. Suppose (3.11)

F mod ([n]q ) ≥ a(n) nq

3.2 Modular Sidon sets. Instead of working with integer arithmetic, it will be more convenient to work holds with overwhelming probability. Then, with overin the cyclic group of order n. A set A ⊂ [n] will be whelming probability, called a mod-n-Sidon set if all the sums (a1 +a2 ) mod n, F mod ([n]p ) with a1 ≤ a2 and a1 , a2 ∈ A, are distinct. For R ⊂ [n], ≥ (1 + o(1))a(n). np we let Fnmod (R) = max |S|, where the maximum is taken

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Finally, choose c so that 4 The upper bounds in Theorems 2.2–2.4 4.1 The key upper bound lemma. The main (4.18) 3 (c/ω) > 3(1/3 + η) and c > 12ω/2(1+3η)(1−2/ω) . technical lemma in the proofs of the upper bounds in Theorems 2.2–2.4 concerns the number of mod-n-Sidon We shall now prove that we may take c0 = c. 2
1/3 sets contained in [n]. Set t = c n log n2 p3 , s = t/ω, σ = Definition 4.1. Suppose t = t(n) is an integer and let (n2 p3 )1/3+η /s, and ξ = 12ω/c2(1+3η)(1−2/ω) . Note that B(t) denote the family of all mod-n-Sidon sets of size t (4.19) t ≥ c(n log 8)1/3 ≥ cn1/3 . contained in [n]. Next we observe that condition (4.12) holds, and hence This is our main lemma. Corollary 4.1 gives that Lemma 4.1. Let t, s = t/ω and σ be such that 3

(4.12)

ω ≥ 4,

0<σ<1

and

2 log(σs) s ≥ . n 1−σ

Then (4.13)

|B(t)| ≤



t 6ωn . t2−2/ω σ 1−2/ω

(4.20)
P [n]p contains a mod-n-Sidon subset of size t t  6ωnp . ≤ t(tσ)1−2/ω Making use of (4.19) and the fact that tσ = ωsσ = ω(n2 p3 )1/3+η , we see that the upper bound in (4.20) is at most  t 6ωnp (4.21) cn1/3 ω 1−2/ω (n2 p3 )(1/3+η)(1−2/ω)  2/ω t 6ω ≤ (n2 p3 )1/3−(1/3+η)(1−2/ω) , c

We give the proof of Lemma 4.1 in Section 5. Very roughly speaking, we shall encode some relevant arithmetic information using certain Cayley-type graphs, and, using certain methods developed for counting graphs with a specified number of vertices and edges that contain no 4-cycles, we shall obtain upper estimates for |B(t)|. which, by (4.17) and the fact that p ≥ 2n−2/3 , is at Lemma 4.1 above will be used in the form of the most following immediate corollary.  t 6ω 3 1/3−(1/3+η)(1−2/ω) (2 ) Corollary 4.1. Suppose t, s, ω and σ are as in (4.22) c Lemma 4.1 and suppose 0 ≤ p = p(n) ≤ 1. Then  t 12ω
≤ = ξt. (4.14) P [n]p contains a mod-n-Sidon set of size t c2(1+3η)(1−2/ω)  t 6ωnp To complete the proof, it suffices to note that (4.18) ≤ |B(t)|pt ≤ 2−2/ω 1−2/ω . t σ implies that ξ < 1. 

4.2 Proof of the upper bound in Theorems 2.2. The upper bound in (2.5) in Theorem 2.2 follows We first prove a binomial, modulo n version of the upper from Lemma 3.1, Corollary 3.1, and Theorem 4.1. bound in Theorem 2.2. 4.3 Proof of the upper bound in Theorem 2.3. Theorem 4.1. For any 0 < δ < 1/3, there is a positive We prove the following result, which, again by constant c02 such that if 2n−2/3 ≤ p = p(n) ≤ n−1/3−δ , Lemma 3.1 and Corollary 3.1, suffices to establish the then, with overwhelming probability, we have upper bound in (2.7) in Theorem 2.3.
1/3 (4.15) F mod ([n]p ) ≤ c02 n log n2 p3 . Theorem 4.2. Suppose 1 ≤ α = α(n) ≤ n1/3 , and let 1 Proof. Let 0 < δ < 1/3 be given. Choose η > 0 small (4.23) p = p(n) = n−1/3 (log n)2/3 . α enough so that Then, with overwhelming probability, we have (4.16) (1 − 3δ)(1/3 + η) < 1/3. log n (4.24) F mod ([n]p ) ≤ c04 (n log n)1/3 , Choose ω ≥ 4 so that log(1 + α) (4.17)

(1/3 + η)(1 − 2/ω) > 1/3.

where c04 is a positive absolute constant.

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Proof. We again use Corollary 4.1. that result with p = p(n) given in (n log n)1/3 and t = ωs, where  log n 1+ (4.25) ω=9 log(1 + α)

In order to apply 1 + α and hence (4.23), we let s = 1+α 1+α t2/ω ≤ = log(1 + α) αω αω 9(log n)(1 + α)  1 log(1 + α) log(n1/3 ) . = ≤ (1 + o(1)) α 9 log n 9 log n 1 1 Also, we take σ = 1/4. = < , 27 + o(1) 24 Before using Corollary 4.1, we first check condition (4.12) holds for all large enough n: if n is large enough, as α ≤ n1/3 . Combining (4.27) and (4.29) implies (4.26), which • ω ≥ 4 holds because completes the proof of Theorem 4.2.  log n log n ≥9 ω ≥ 9 log(1 + α) log(1 + n1/3 ) 4.4 Proof of the upper bound in Theorem 2.4. In view of Lemma 3.1 and Corollary 3.1, it suffices to log n = 27 + o(1) ≥ 4. = (9 + o(1)) 1/3 prove the following result. log n • σ < 1 holds by the definition of σ = 1/4. s3 2 log(σs) • ≥ holds since n 1−σ

Theorem 4.3. If p = p(n) ≤ n−1/3 (log n)5/3 , then (4.30)

holds with overwhelming probability. Furthermore, if

3

s n log n ≥ = log n, n n and 2 log(σs) 1−σ

=

1  8 log (n log n)1/3 3 4 8 1 8 · log n = log n. 3 3 9

F mod ([n]p ) ≤ 6n1/3 (log n)4/3

(4.31)

p = p(n) ≥ n−1/3 (log n)5/3 ,

then, with overwhelming probability, p (4.32) F mod ([n]p ) ≤ 6 np log n.

Proof. The right-hand side of (4.32) is equal to the right-hand side of (4.30) when p = n−1/3 (log n)5/3 . Hence, by Fact 3.1, it is enough to prove that (4.32) holds with overwhelming probability for all p as Since the condition of Lemma 4.1 is satisfied, the in (4.31). To that end, let p = p(n) as in (4.31) be only thing we shall check for the proof of Theorem 4.2 given. √ is whether the base of the exponential in (4.13) is less Set ω = log n, σ = 1/2 and t = 6 np log n. than 1. That is, whether With this choice of parameters, the first two inequalities in (4.12) are clearly satisfied. Recalling (4.31), we see 6ωnp 6 ωnp (4.26) = 1−2/ω 2−2/ω < 1. that the third inequality in (4.12) also holds. 2−2/ω 1−2/ω t σ σ t Having verified (4.12), we invoke Corollary 4.1 and Clearly, obtain that the probability that [n]p should contain a mod-n-Sidon set of t elements is at most (4.27) 6/σ 1−2/ω ≤ 6/σ = 24.  t 6(log n)np √ Note that (6 np log n)2−2/ω (1/2)1−2/ω √  t ωnp ωn1−1/3 (log n)2/3 /α 1 (6 np log n)2/ω 1−2/ω (4.28) = = . √ ≤ 6(log n)np 2 t2 αω ω 2 n2/3 (log n)2/3 (6 np log n)2 (4.33) t  (62 np log n)1/ω We claim that ≤ 12np log n 62 np log n t2/ω 1 ωnp t  t  (4.29) = < . 12(62 n log n)1/ log n 33 αω 24 t2−2/ω ≤ ≤ , 2 6 62 Indeed, condition (4.29) is fulfilled with the choice (4.25)  of ω: as ω ≥ 2(log n)/ log(1+α), we have t2/ω ≤ n2/ω ≤ completing our proof. ∼

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Fact 5.2. A set A ⊂ [n] contains the solu5 Proof of Lemma 4.1 5.1 Solutions. We start with the following defini- tion (a1 , a2 , b1 , b2 ) of the equation (5.34) if and only if the graph GA contains the walk (0, a1 b1 a2 b2 ) and this tion. walk is closed. Definition 5.1. (a) A solution of the equation In view of the above fact, we introduce the following (5.34) x1 + x2 = y1 + y2 (mod n) terminology. is a quadruplet (a1 , a2 , b1 , b2 ) ∈ [n]4 with a1 + a2 = Definition 5.4. A closed walk (0, a1 b1 a2 b2 ) b1 + b2 (mod n). in GA is nontrivial if the corresponding solution (b) A set A ⊂ [n] contains a solution (a1 , a2 , b1 , b2 ) (a1 , a2 , b1 , b2 ) of (5.34) is nontrivial. Otherwise, we say that (0, a1 b1 a2 b2 ) is a trivial closed walk. if a1 , a2 , b1 and b2 are in A. The following simple remark gives equivalent con- Definition 5.5. We call a walk in GA a rainbow walk if all the edges in the walk have distinct labels. ditions for a set A ⊂ [n] to be a mod-n-Sidon set. Fact 5.1. Let A ⊂ [n] be given. The following are equivalent. (i) A ⊂ [n] is a mod-n-Sidon set. (ii) (iii)

A does not contain any solution of (5.34) with {a1 , a2 } 6= {b1 , b2 }.

(a1 , a2 , b1 , b2 )

A does not contain any solution (a1 , a2 , b1 , b2 ) of (5.34) with {a1 , a2 } ∩ {b1 , b2 } = ∅.

We have the following fact. Fact 5.3. A set A ⊂ [n] is a mod-n-Sidon set if and only if every closed walk of length 4 that GA contains is trivial. The following fact gives an equivalent condition for a closed walk (0, a1 b1 a2 b2 ) to be nontrivial.

Fact 5.4. Let W = (0, a1 b1 a2 b2 ) be a closed walk of In view of the above fact, we introduce the following length 4 in GA . The walk W is nontrivial if and only if definition. both b1 = 6 a1 and b1 6= a2 . Definition 5.2. A solution (a1 , a2 , b1 , b2 ) of (5.34) is The proof of Lemma 4.1 relies on the following called a nontrivial solution if it satisfies the second (or proposition. equivalently third) condition in Fact 5.1. Proposition 5.1. Let A ⊂ [n] be a mod-n-Sidon set 5.2 A Cayley-type graph. Our approach for count- and let v ∈ [n] \ A be given. If v is connected in GA ing the number of mod-n-Sidon sets of a given cardinal- to an element of A by a rainbow walk of length 2 in ity contained [n] is based on encoding some arithmetic which the middle vertex is nonzero, then A ∪ {v} is not information in some graphs. It will be convenient to a mod-n-Sidon set. consider edge-labelled graphs, which may have loops, but will have no multiple edges. Proof. Suppose v is connected to u ∈ A by a rainbow walk W of length 2 in GA and suppose the middle vertex Definition 5.3. For a set A ⊂ [n], we define GA as the w of W is nonzero. By Fact 5.3, it is enough to show edge-labelled graph with vertex set [n] in which vertex u that the graph G A∪{v} contains a nontrivial closed walk is adjacent to vertex v if (u + v) mod n ∈ A. The of length 4. edge {u, v} will be labelled with the label (u + v) mod n. Since u and v are in A ∪ {v}, the graph GA∪{v} contains the edges {0, u} and {0, v}. These edges Note that GA will have a loop labelled 2u mod n at together with the edges {u, w} and {w, v} in W form the the vertex u if 2u mod n ∈ A. closed walk (0, u(u + w)(v + w)v) of length 4 in GA∪{v} . Our next observation states that a solution of The corresponding solution is u + (v + w) = (u + w) + v. equation (5.34) contained in A can be regarded as a Since w = 6 0 and u 6= v, the solution is nontrivial. closed walk of length 4 in GA and vice versa. We will Hence the closed walk (0, u(u + w)(v + w)v) in GA∪{v} use the following notation for walks in GA . is nontrivial. This completes the proof.  Notation 5.1. Let v ∈ [n] and ai ∈ A (1 ≤ i ≤ l) be given. The unique walk of length l in GA starting Remark 5.1. The converse of Proposition 5.1 also at v and whose ith edge is labelled ai (1 ≤ i ≤ l) is holds, but this fact will not be required in the proof of Lemma 4.1. denoted (v, a1 a2 . . . al ).

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5.3 Proof of Lemma 4.1. Lemma 4.1 gives an upper bound on the cardinality of the family of modn-Sidon sets of size t. The idea of the proof, based on a method introduced in [14] (see also [8, 16]), is as follows. For a suitably chosen s < t, each mod-n-Sidon set of size t is generated in two phases: (i) The first s elements are chosen arbitrarily to form a mod-n-Sidon set B0 .

(Forbidden)

(Forbidden)

(ii) The remaining t − s elements are chosen by Algorithm 5.1, given below. Note that, in general, there will be many ways of extending the set B0 from phase (i) in phase (ii). To reflect this fact, Algorithm 5.1 will be a non-deterministic algorithm. To spell it out: certain steps of the algorithm will involve arbitrary choices; corresponding to each sequence of choices we shall have a specific output set of t − 2s elements, extending B0 . We will show that, when the first s elements of phase (ii) are chosen, the set of all “candidates” for the remaining t − 2s elements is very small, and this will let us prove Lemma 4.1. The logical structure of the proof of Lemma 4.1, which is based on two propositions and three lemmas, is as in the following diagram. Lem 5.2 =⇒ Cor 5.1 =⇒ Lem 5.1 =⇒ Lem 4.1 ⇑ Proposition 5.2 ⇑ Proposition 5.1

Figure 2: Algorithm 5.1 (case i = 0) in GB0 to v by a rainbow walk of length 2 in which the middle vertex is nonzero. Let G2B0 be the graph on the vertex set [n], with u adjacent to v if and only if u ∼ v. Definition 5.6. Let G be a graph and suppose C ⊂ V (G) is given. As usual, the subgraph of G induced by C is denoted G[C]. An ordering x1 , . . . , x|C| of the vertices in C is called a G-maxdeg order, or simply a maxdeg order, of C if, for all 1 ≤ i ≤ |C|, the vertex xi has maximum degree in the subgraph G[{xj : i ≤ j ≤ |C|}] of G induced by the vertices xj (i ≤ j ≤ |C|). We may now describe our algorithm, Algorithm 5.1. The reader may find Figure 2 of some help while reading this algorithm. As mentioned before, the reason we are interested in this algorithm is given in Proposition 5.2 (see also the paragraph before Fact 5.5).

The lemmas and propositions above are proven under Algorithm 5.1. With input an integer t and a modthe assumption that n-Sidon set B0 ⊂ [n] with |B0 | = s < t, this algorithm outputs, non-deterministically, when it does not fail, a the initial set B0 of the first s elements (5.35) set A = {a1 , . . . , at−s } ⊂ [n] \ B0 . We shall show that is a mod-n-Sidon set. the family A(t, B0 ) of all possible outputs A contains the family of all sets A0 ⊂ [n] \ B0 with |A0 | = t − s such Before we proceed, we mention a technical point: 0 Algorithm 5.1 may output some ‘bad’ sets A, that is, that A ∪ B0 is a mod-n-Sidon set. The algorithm proceeds as follows. such that B ∪ A is not a mod-n-Sidon. However, we 0

shall show in Proposition 5.2 that any set A with B0 ∪A a mod-n-Sidon can be generated by that algorithm, by appropriate non-deterministic choices during its execution. We shall show that the number of possible outputs of Algorithm 5.1 is suitably bounded from above, whence our bound on the number of mod-n-Sidon sets will follow. We now introduce some notation. Notation 5.2. Let B0 ⊂ [n] be given. For u and v ∈ [n], we write u ∼ v, or simply u ∼ v, if u is connected B0

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1. Set C0 = [n] \ B0 . 2. For 0 ≤ i ≤ t − s suppose we have already defined the “candidate sets” C0 ⊃ C1 ⊃ · · · ⊃ Ci and a1 , . . . , ai ∈ A, with aj ∈ Cj−1 for all 1 ≤ j ≤ i. 3. If i = t − s, output A = {ai : 1 ≤ i ≤ t − s} and stop. 4. If i < t − s and Ci = ∅, then the algorithm fails and stops, and there is no output.

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5. Suppose now that i < t − s and that Ci 6= ∅. Let y1 , . . . , y|Ci | be the vertices of G2B0 [Ci ] listed in a maxdeg order. Pick li+1 with 1 ≤ li+1 ≤ |Ci | arbitrarily, and set ai+1 = yli+1 . (This is the key non-deterministic step in the algorithm.)

Fact 5.5. Let t > 2s. Suppose Algorithm 5.1 has produced the output A = {a1 , . . . , at−s } and let C0 ⊃ · · · ⊃ Ct−s be the sets Ci built in this execution of that algorithm. Let A∗ = {a1 , . . . , as }. Then every element in A \ A∗ = {as+1 , . . . , at−s } belongs to Cs .

6. Set Li = {yj : 1 ≤ j ≤ li+1 } and let Ri We shall show that, if s is suitably large, then |Cs | be the neighbourhood of yli+1 in the graph is small, and hence A \ A∗ is confined to a small set. G2B0 [{yj : li+1 ≤ j ≤ |Ci |}]. This will imply that |A(t, B0 )| cannot be too large. 7. Let Ci+1 = Ci \ (Li ∪ Ri ) and let i ← i + 1. Go to Lemma 5.1. Suppose 0 < σ < 1 and s are such that Step 2. Note that, in Step 5, we have freedom in the choice 2 log(σs) s3 ≥ . (5.36) of ai+1 , reflecting the fact that there are possibly many n 1−σ sets A with B0 ∪ A a mod-n-Sidon set. Let A(t, B0 ) be the family of all the possible outputs of Algorithm 5.1 Then when it is run on input (t, B0 ). n . (5.37) |Cs | ≤ σs Proposition 5.2. Let t and B0 form an input to Algorithm 5.1. If A0 ⊂ [n] \ B0 with |A0 | = t − s is such We shall prove Lemma 5.1 in Section 5.4. that B0 ∪ A0 is a mod-n-Sidon set, then A0 ∈ A(t, B0 ) Lemma 4.1 follows easily from Lemma 5.1. Proof. Let A0 as in the statement of the lemma be given. It suffices to show that there exist suitable non- Proof. [Proof of Lemma 4.1 (assuming Lemma 5.1)] deterministic choices li+1 in Step 5. Let us give an Recall that B(t) denotes the family of all mod-n-Sidon inductive proof of this fact. Suppose 0 ≤ i < t − s, and sets of cardinality t contained in [n]. Also, recall that, assume that the algorithm has already determined the by the assumptions in Lemma 4.1, we have s = t/ω sets C0 ⊃ C1 ⊃ · · · ⊃ Ci and the vertices a1 , . . . , ai ∈ and σ with A0 , with aj ∈ Cj−1 for all 1 ≤ j ≤ i. Furthermore, s3 2 log(σs) ω ≥ 4, 0 < σ < 1 and ≥ . suppose that, when making the non-deterministic choice n 1−σ in Step 5 to choose lj+1 (0 ≤ j < i), we have chosen   the smallest integer l with 1 ≤ l ≤ |Cj | satisfying [n] For every mod-n-Sidon set B0 ∈ , consider the that yl ∈ A0 . s   The crucial remark is that every vertex in A0 \ [n] − B0 sets A = {a1 , . . . , at−s } ∈ such that B0 ∪ {a1 , . . . , ai } must be in t−s [ [ A ∈ B(t). In order to count the number of such sets A, Ci := C0 − Lj − Rj . we first note that the number of possibilities for the 0≤j
of Lemma 4.1, it follows from Lemma 5.1 that |Cs | ≤ n/σs. set   Consequently, for every mod-n-Sidon   B0 ∈ [n] [n] − B0 , the number of sets A ∈ such s   t − s n n/σs that B0 ∪ A ∈ B(t) is at most . Hence s t − 2s

166

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we infer that (5.38) n |B(t)| ≤ (B0 , A) : B0 ∪ A ∈ B(t) where B0 ∈ B(s) and A ∈     n n n/σs ≤ . s s t − 2s

[n]−B0 t−s


o ei Figure 3: Definition of C

Inequality (5.38) with s = t/ω yields that  2      t−2s n n/σs en 2s en |B(t)| ≤ ≤ s t − 2s s σs(t − 2s)  en 2s  en t−2s  t−2s 1 = s s σ(t − 2s) t(1−2/ω)  en t  1 = s σt(1 − 2/ω)  eωn t  t(1−2/ω) 1 = t σt(1 − 2/ω) t  n = C 2−2/ω 1−2/ω , t σ

Lemma 5.2. Let 0 ≤ i < s. If n , σs

ei | > |C then we have |Ci+1 | ≤ |Ci | exp

n

−

1 − σ s2 o . 2 n

e0 ⊃ C1 ⊃ C e1 ⊃ · · · ⊃ C es−1 ⊃ Cs , Since C0 ⊃ C Lemma 5.2 yields the following.

where C=

The proof of Lemma 5.1 is based on Lemma 5.2 below.

eω eω 2−2/ω = . (1 − 2/ω)1−2/ω (ω − 2)1−2/ω

Corollary 5.1. Either

Note that if ω ≥ 4, then ω − 2 ≥ ω/2, and hence C ≤ eω21−2/ω < 6ω, which completes the proof of Lemma 4.1. 

|Cs | ≤

n σs

or

In order to complete the proof of Lemma 4.1, it remains to prove Lemma 5.1.

n 1 − σ s3 o h n 1 − σ s2 ois = n exp − |Cs | ≤ n exp − 2 n 2 n

5.4 Proof of Lemma 5.1. In this section we will holds. Consequently prove Lemma 5.1. For notational convenience we first n n 1 − σ s3 o adjust some of the notation and terminology used in (5.41) |C | ≤ max , n exp − . s Algorithm 5.1. Let us first recall that, in Step 5 in that σs 2 n 2 algorithm, we considered the vertices of GB0 [Ci ] listed Now Lemma 5.1 follows from Corollary 5.1. in a maxdeg order, namely, y1 , . . . , y|Ci | . Notation 5.3. For i = 1, 2, . . . , s − 1, consider Ci with Proof. [Proof of Lemma 5.1] Since the assumption its associated labeling {y1 , . . . , yci } where ci = |Ci |. 2 log(σs) s3 (a) Let l = li+1 be the label satisfying yl = ai+1 . ≥ n 1−σ e (b) Set Ci = Ci − {y1 , . . . , yl−1 }, and let Ri (yj ), where can be rewritten as 1 ≤ j ≤ ci , denote the neighbourhood of yj in the n 1 − σ s3 o graph G2B0 [Ci − {y1 , . . . , yj−1 }]. Note that, clearly, n ≥ n exp − , σs 2 n ei ⊃ Ri (yl ). (5.39) C inequality (5.41) implies that |Cs | ≤ n/σs, which (c) The definition of Ci+1 in Step 7 of Algorithm 5.1 completes the proof of Lemma 5.1.  can be rewritten as   The rest of this section is devoted to the proof of ei \ {yl } ∪ Ri (yl ) ⊂ C ei \ Ri (yl ). (5.40) Ci+1 = C Lemma 5.2.

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To get a lower bound on |Ri (yl )|, we estie Define the auxiliary bipartite graph G b= mate |E(G)|. b b e e GB0 as follows. Set V (G) = Ci ∪ [n]. (For u ∈ Ci , conei and the other in [n].) sider two copies of u: one in C ei and v ∈ [n], if {y, v} ∈ E(GB ), then we add For y ∈ C 0 ei × [n] to E(G). b the edge (y, v) ∈ C e we consider the set of In order to estimate |E(G)|, b 2-paths in G: P=

n

b (y1 , v), (y2 , v) : (y1 , v), (y2 , v) ∈ E(G) o ei and v ∈ [n] . where y1 6= y2 ∈ C

Note that

e and G b Figure 4: The graphs G

X deg b (v) G (5.44) |P| = . ei and Ri (yl ) be as in Proof. [Proof of Lemma 5.2] Let l, C 2 v∈[n] Notation 5.3. We shall show that, under the assumption ei | > n/σs, we have |C We claim that 1 − σ s2 e |Ci |. (5.42) |Ri (yl )| ≥ e = |P|. (5.45) |E(G)| 2 n This together with (5.39) and (5.40) yields that  2 ei | − |Ri (yl )| ≤ 1 − 1 − σ s |C ei | |Ci+1 | ≤ |C 2 n  1 − σ s2  ≤ 1− |Ci | 2 n n 1 − σ s2 o ≤ |Ci | exp − , 2 n which completes the proof of Lemma 5.2.   e = G2 C ei , where It remains to prove (5.42). Let G B0   2 ei is defined in Notation 5.2 and Definition 5.6. In GB0 C ei , we have that {y, y 0 } ∈ E(G) e other words, for y, y 0 ∈ C 0 if and only if y is connected to y by a rainbow walk of length 2 in GB0 in which the middle vertex is nonzero. Note that middle vertices of rainbow walks may be ei . (See Figure 4.) Owing to the definition outside C of Ri (y) as in Notation 5.3(b), we observe that X e = |E(G)| |Ri (y)|. ei y∈C

e ≤ |P|. To check this, let We first observe that |E(G)| e e the vertices y1 {y1 , y2 } ∈ E(G). By the definition of G, and y2 are connected by a rainbow walk of length 2 in GB0 , with the middle vertex different from 0. Since this walk is a rainbow walk, we have y1 6= y2 . Also, this rainbow walk can be broken to two edges (y1 , v), (y2 , v) ∈  b where v ∈ [n] \ {0}, and hence (y1 , v), (y2 , v) ∈ E(G), e < P. Now suppose for a contradiction that |E(G)| e where y1 6= y2 , |P|. For an edge {y1 , y2 } ∈ E(G), there exist distinct vertices v , v 1 2 ∈ [n] \ {0} such that both (y1 , v1 ), (y2 , v1 ) and (y1 , v2 ), (y2 , v2 ) are in P. However, if this happens, GB0 contains a closed walk of length 4, passing through the vertices y1 , v1 , y2 and
v2 , which is y1 , (y1 + v1 )(v1 + y2 )(y2 + v2 )(v2 + y1 ) . By Fact 5.4, this closed walk is nontrivial. Hence Fact 5.3 implies that B0 is not a mod-n-Sidon set, contradicting the assumption on B0 in (5.35). Consequently, our claim (5.45) holds. Combining (5.44) and (5.45) yields that

By Notation 5.3(a) and (b), yl is the first element e = |P| = (5.46) |E(G)| ei . Consequently, by the definition of the labeling of in C ei in Algorithm 5.1, we clearly have C |Ri (yl )| = max |Ri (y)|.

X deg b (v) G . 2

v∈[n] x 2




By the convexity of f (x) =

, we infer that

ei y∈C

P

Hence we bound |Ri (yl )| from below as P e ei |Ri (y)| |E(G)| y∈C (5.43) |Ri (yl )| ≥ = . ei | ei | |C |C

e |E(G)|

168

 degGb (v)/n 2  P deg ei b (y)/n y∈C G ≥ n . 2 ≥ n

v∈[n]

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Since degGb (y) = degGB0 (y) = |B0 | = s, we have

• The vertex set of S is [n].

• For every nontrivial solution (a1 , . . . , al ) of (5.34) such that the ai ’s are distinct elements of [n], we add the hyperedge {a1 , . . . , al } to S.   ei | > n/(σs), we have Let S [n]m be the sub-hypergraph of S induced on Owing to our assumption |C [n]m . Let D be a set of vertices obtained by choosing   e an arbitrary vertex from each hyperedge of S [n]m . s|Ci | 1<σ , Observe that [n]m \ D is an independent set of S [n]m , n and hence [n]m \ D is a mod-n-Sidon subset of [n]m . and hence, from (5.47), we have Consequently, ei | ei |2 ei | F mod ([n]m ) ≥ [n]m \ D = [n]m − |D| s|C 1 − σ s2 |C s|C e · (1 − σ) ≥ . |E(G)| >   2 n 2 n (7.49) ≥ [n]m − S [n]m   = m − S [n]m . Consequently, in view of (5.43), the last inequality   yields that S [n]m . We first have that the Now we estimate h  i  e 1 − σ s2 e |E(G)| expectation E S [n]m is ≥ |Ci |, (5.48) |Ri (yl )| ≥ ei | 2 n |C h  i  i X h E S [n]m = P E ⊂ [n]m which is equivalent to (5.42), and hence this completes E∈S the proof of Lemma 5.2.  X  n − 4  n −1 = m−4 m 6 Nontrivial solutions in random sets E∈S  m 4 X Based on Definition 5.2, recall that a solu≤ . n tion (a1 , a2 , b1 , b2 ) of (5.34) is called a nontrivial E∈S solution if it satisfies the second (or equivalently third) condition in Fact 5.1. Let us define a random variable Owing to |S| = O(n3 ) and the assumption m  n1/3 , we have that will be important for us. h   m4   i = o(m). E S [n]m = O Definition 6.1. Let X = X([n]p ) denote the number n of nontrivial solutions (a1 , a2 , b1 , b2 ) of (5.34) contained Markov’s inequality implies that we almost surely have in [n]p .   S [n]m = o(m). (7.50) Lemma 6.1 below gives an estimate for X required in the proofs of the lower bounds in Theorems 2.2–2.4. Given (7.50), combining (7.49) with F mod ([n]m ) ≤ m yields that we almost surely have F mod ([n]m ) = Lemma 6.1. Fix 0 < δ < 1/12 and suppose p ≥ m(1 − o(m)).  n−3/4+δ . Then, with overwhelming probability, we have X = n3 p4 (1 + o(1)). 8 The lower bounds in Theorems 2.2–2.4 e ≥n (5.47) |E(G)|

 e   ei | ei |  s|C s|Ci |/n 1 s|C = n· −1 . 2 2 n n

Lemma 6.1 may be proved by applying a concentra- The lower bounds in Theorems 2.2–2.4 are obtained combining two lower bounds, given separately below. tion result due to Kim and Vu [13]. The first lower bound relies on a result on independent sets in hypergraphs. Before stating the result, 7 Proof of Theorem 2.1 we introduce some definitions. A hypergraph is called Because of Lemma 3.2, the trivial bound F ([n]m ) ≤ m simple if any two of its hyperedges share at most one and the following result imply (2.4) in Theorem 2.1. vertex. A hypergraph is r-uniform if all its hyperedges have cardinality r. A set of vertices I of a hypergraph 1/3 Theorem 7.1. For m  n , we almost surely have is independent if it contains no hyperedge of the hypermod graph. We shall use the following extension of a celeF ([n]m ) = m(1 − o(m)). brated result due to Ajtai, Koml´os, Pintz, Spencer and Proof. We use the usual deletion method. Let S be the Szemer´edi [1], obtained by Duke, Lefmann and the third author [5]. hypergraph defined as follows:

169

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Theorem 8.1. Let H be a simple r-uniform hypergraph with N vertices and average degree at most tr−1 for some t. Then H has an independent set of size at least (8.51)

This proves the lower bound in (2.9) in Theorem 2.4. References

(log t)1/(r−1) N, c t

where c = c(r) is a positive constant that depends only on r. Combining Lemma 6.1 and Theorem 8.1, one may prove the following result, which is the binomial, modulo n version of the lower bound in Theorem 2.2. Theorem 8.2. There exists an absolute constant d > 0 such that, for p ≥ 2n−2/3 , we have
1/3 (8.52) F mod ([n]p ) ≥ d n log(n2 p3 ) with overwhelming probability. Based on Corollary 3.1 and Lemma 3.1, we obtain a uniform, non-modular version of Theorem 8.2. Corollary 8.1. (First lower bound) There exists an absolute constant d > 0 such that, for m ≥ 2n1/3 , we have
1/3 (8.53) F ([n]m ) ≥ d n log(m3 /n) with overwhelming probability. Corollary 8.1 gives the lower bound in (2.5) and in (2.7) in Theorems 2.2 and 2.3. We now consider a different argument. For larger m = m(n), it turns out that, instead of using Theorem 8.1, it is better to make use of a result by Koml´os, Sulyok, and Szemer´edi [18]. For a set A of integers, recall that F (A) is the maximum size of a Sidon set contained in A. Theorem 8.3. There is a positive absolute constant d0 such that, for all sufficiently large m and all A with |A| = m, we have F (A) ≥ d0 F ([m]). Together with the Chowla–Erd˝ os [3, 6] result, which implies that √ F ([m]) ≥ (1 + o(1)) m, Theorem 8.3 yields the following result. Theorem 8.4. (Second lower bound) There is an absolute constant d0 > 0 such that, for m = m(n)  1, we have √ F ([n]m ) ≥ d0 m with probability one.

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