LOWER BOUNDS FOR THE TUR ´AN DENSITIES OF DAISIES

. For integers r ≥ 3 and t ≥ 2, an r -uniform t -daisy D tr is a family of (cid:0) 2 tt (cid:1) r -element sets of the form { S ∪ T : T ⊂ U, | T | = t } for some sets S, U with | S | = r − t , | U | = 2 t and S ∩ U = ∅ . It was conjectured by Bollob´as, Leader and Malvenuto in [2] (and independently by Bukh) that the Tur´an densities of t -daisies satisfy lim r →∞ π ( D t r ) = 0 for all t ≥ 2; this has become a well-known problem, and it is still open for all values of t . In this paper, we give lower bounds for the Tur´an densities of r -uniform t -daisies. To do so, we introduce (and make some progress on) the following natural problem in additive combinatorics: for integers m ≥ 2 t ≥ 4, what is the maximum cardinality g ( m,t ) of a subset R of Z /m Z such that for any x ∈ Z /m Z and any 2 t -element subset X of Z /m Z , there are t distinct elements of X whose sum is not in the translate x + R ? This is a slice-analogue of the extremal Hilbert cube problem considered in [3] and [4].


Introduction
For integers r ≥ 3 and t ≥ 2, an r-uniform t-daisy D t r is a collection of 2t t r-element sets of the form {S ∪ T : T ⊂ U, |T | = t} for some sets S, U with |S| = r − t, |U | = 2t and S ∩ U = ∅. As usual, for an integer n ≥ 3 we write ex(n, D t r ) for the nth Turán number of D t r , i.e. the maximum possible cardinality of a family of r-element subsets of {1, 2, . . . , n} which is D t r -free, and we write π(D t r ) = lim n→∞ ex(n, D t r ) n r for the Turán density of D t r . In the case r = 2, we have π(D 2 2 ) = 2/3 by Turán's theorem for K 4 s. The first unknown case occurs when r = 3; in this case, Bollobás, Leader and Malvenuto show in [2] that π(D 2 3 ) ≥ 1 2 (by taking the complement of the Fano plane, blowing up and iterating) and conjecture that in fact equality holds. For larger t ≥ 2 and r ≥ 3, even less is known concerning π(D t r ). The following conjecture was made by Bollobás, Leader and Malvenuto in [2] (and independently by Bukh, see [2]).
This is still open even for t = 2. The value of π(D t r ) is clearly nondecreasing in t, since an r-uniform family that is free of t-daisies is also free of t ′ -daisies for all t ′ > t. In [2] a lower bound of π(D 2 r ) ≥ r!/r r is observed, which comes from considering the r-partite r-uniform hypergraph on [n] with parts of sizes as equal as possible. However, this lower bound is exponentially small in r. In this paper, we obtain the following improved lower bound (which is polynomial in r), using an additive-combinatorial construction. We also raise a question in additive combinatorics which may be of interest in its own right. Theorem 1.2. There exists an absolute constant c > 0 such that π(D 2 r ) ≥ c/r for all r ≥ 3. Furthermore, for each t ≥ 3, we have .
Our proof of Theorem 1.2 relies upon the existence of a subset of Z m := Z/mZ that avoids a certain additive structure, which we define now. Definition 1.3. For positive integers m, t ≥ 2 with m ≥ 2t, let g(m, t) denote the maximum possible size of a subset R ⊂ Z m such that for any x 0 ∈ Z m and any (2t)-element subset X of Z m , there are t distinct elements of X whose sum is not contained in R − x 0 , i.e.
For brevity, given a set X ∈ Zm 2t we write for the set of sums of t distinct elements of X; g(m, t) is the maximum size of a subset of Z m containing no translate of C(X) for any |X| = 2t.
The function g(m, t) is related to a question raised by Gunderson and Rödl in [4], concerning Hilbert cubes.
Gunderson and Rödl considered large sets of integers which do not contain any translate of a Hilbert cube (working over the integers, i.e., the R = Z case of Definition 1.4). In particular, they prove the following (Theorem 2.3 and Theorem 2.5 of [4] that does not contain any translate of a Hilbert cube. (Here, and henceforth, we write [m] := {1, 2, . . . , m} for the standard m-element set.) We consider the case R = Z m , but for our purposes, the structural differences between [m] and Z m will not be particularly important. Estimating g(m, t) is a natural variant of the Gunderson-Rödl problem, where we avoid only the middle slice of a Hilbert cube of dimension 2t. We make a small but important change in that, in Definition 1.3, we require that X be composed of 2t distinct elements, while a Hilbert cube may even have (In this case, the Hilbert cube is a d-element arithmetic progression, but C(X) is a singleton.) We obtain the following bounds on g(m, t).
, and if furthermore m is prime, then .
For t = 2 and m ≥ 64 we have g(m, 2) ≥ √ m/8. Theorem 1.7. For each t ≥ 2 and all m sufficiently large depending on t, we have (Here, we use the standard asymptotic notation: if X is a set and f, h : It would be of interest to narrow the gap between our upper and lower bounds on g(m, t).
The proofs of Theorem 1.7 and of the first part of Theorem 1.6 (i.e. the t > 2 case) are very similar to those used in [4] to prove Theorem 1.5 (above). Our lower bound for t > 2 consists of a probabilistic construction very similar to that of Gunderson and Rödl in [4]; we have to choose a random set of slightly lower density as we must avoid the middle slice of a Hilbert cube, as opposed to an entire Hilbert cube. At first sight, it might seem that an upper bound on g(m, t) follows from the upper bound given in Theorem 1.5, since a set which contains a Hilbert cube contains its middle layer, but one must make small changes to the proof in [4] so as to ensure that the generators of the Hilbert cube we find, the x i , are distinct. We give an explicit construction for the t = 2 case in Theorem 1.6: this construction outperforms the probabilistic one in the case t = 2. The paper is structured as follows. In Section 2 we prove Theorem 1.2 using Theorem 1.6, and in Section 3 we prove Theorems 1.6 and 1.7 (note that we do not require the latter in our study of the Turán density of daisies, but it may be of independent interest).

The Proof of Theorem 1.2
Proof. In proving Theorem 1.2, by an appropriate choice of c (and of the absolute constant implicit in the big-Oh notation), we may clearly assume that r ≥ 8. For r, t ∈ N with r ≥ 8 and n sufficiently large depending on r and t, we proceed to construct a D t r -free family of r-element subsets of [n]. We use a 'partite' construction, partitioning [n] into L blocks, and then taking only r-sets containing at most one element from each block. We may assume that n ≥ 2r 2 . Let L be a prime number such that r 2 ≤ L ≤ 2r 2 (such exists, by Bertrand's postulate), and for each i ∈ [n], set x i = L i−1 n ; note that 0 ≤ x i < L for each i ∈ [n]. By Theorem 1.6, there exists a set R ⊂ Z L of size |R| = g(L, t) with the property that for any X ⊂ Z L with |X| = 2t and for any x 0 ∈ Z L , we have r by must satisfy x 0 +C(X) ⊂ R, and therefore there is a t-sum (indexed by T = {i 1 , i 2 , . . . , i t } ⊂ U , say) such that It follows that S := S 0 ∪ T / ∈ F R and therefore D ⊂ F R , as required. Now to finish the proof of Theorem 1.2 we bound |F R | from below. First note that there are at least 1 2 n r sets S ∈ [n] r with x i = x j for all i = j, i, j ∈ S. Indeed, choose a set S uniformly at random from [n] r . Since the probability that a uniformly random two-element subset {i, j} of [n] has x i = x j is at most 1/L, we have The family F = F R , defined above, is D t r -free even if the set R is replaced by a translate R a := R + a for some a ∈ Z L . Averaging over all such translates yields some translate R a of R such that |F Ra | ≥ 1 2 n r |R| L , and therefore Now we may apply Theorem 1.6 (recalling that r 2 ≤ L ≤ 2r 2 is prime): when t = 2 and L = m ≥ 64 (which follows from r ≥ 8), we have and when t > 2 we have , as required.

Bounds on g(m, t)
The focus of this section is the analysis of g(m, t).
Proof. First assume t ≥ 3. In this case, we use the idea of Gunderson and Rödl (in [4]) of passing to a fairly dense subset of Z m which is free of 3-term arithmetic progressions; this 'destroys' a lot of the additive structure we want to avoid. By the well-known construction of Behrend in [1], there exists a set , log denoting the natural logarithm) that contains no 3-term arithmetic progression. Let R 1 ⊂ Z m be the natural embedding of R 0 into Z m . Then R 1 also contains no 3-term arithmetic progression. Set and choose a set R 2 ⊂ R 1 by including each element of R 1 independently at random with probability p. A standard Chernoff bound (for example Theorem 3.5 in [5]) yields (1) P(|R 2 | ≤ |R 1 |p/2) ≤ e −|R 1 |p/8 .

Define the random variable
Since R 2 does not contain any 3-term arithmetic progressions, for any set of the form x 0 +C(X) lying within R 2 , we must have |x 0 +C(X)| = |C(X)| = 2t t . Indeed, suppose for a contradiction that X = {x 1 , . . . , x 2t } is a (2t)-element subset of Z m with x 0 +C(X) ⊂ R 2 , where x 0 ∈ Z m and |C(X)| < 2t t . Then there exist two distinct t-element subsets of X, we may assume without loss of generality that x i 1 ∈ S 1 \ S 2 and x i ′ 1 ∈ S 2 \ S 1 , so that We now proceed to bound EY from above. In the case that m is not prime we may crudely bound the number of possible sets of the form x 0 + C(X) from above by m 2t+1 (which is the number of choices for x 0 , x 1 , . . . , x 2t ∈ Z m ). If m is prime then we may assume each such set has x 0 = 0, by translating each of x 1 , . . . , x 2t by t −1 x 0 , leaving only m 2t choices. The probability that each fixed set of the form It follows from Markov's inequality that Combining (1) and (2), we obtain Clearly, for any t ≥ 2 and m sufficiently large depending on t, we have 1−γ(m)− γ(m)+2t ( 2t t )−1 > 0, so for large enough m, the probability in (3) is positive, and therefore there exists a set R 2 ⊂ Z m with |R 2 | > m 1−γ(m) p/2 and Y < m 1−γ(m) p/4. Now for each set of the form x 0 + C(X) ⊂ R 2 for (x 0 , X) = (x 0 , {x 1 , . . . , x 2t }) we remove a single element from R 2 , chosen arbitrarily from x 0 + C(X). The total number of elements deleted from R 2 is at most Y < m 1−γ(m) p/4 and we are still left with elements, finishing the proof of the first statement of Theorem 1.6. Finally, in the case t = 2, we give an algebraic construction that improves upon the random one. First we recall the definition of a Sidon set. It follows from the classical construction of Singer [6] that for any prime p there is a Sidon set of size p + 1 inside Z p 2 +p+1 . Assume that m ≥ 64 and let p be a prime with √ m/8 ≤ p ≤ √ m/4 (such exists, by Bertrand's postulate). Let R 0 be a Sidon set of size at least √ m/8 inside Z p 2 +p+1 . The image R of R 0 under the natural inclusion map from Z p 2 +p+1 to Z m is a Sidon set in Z m (here we use p 2 +p+1 ≤ m/16+ √ m/4+1 < m/2). Now we will show that for any x 0 and X = {x 1 , x 2 , x 3 , x 4 } ∈ Zm 4 we have x 0 + {x 1 + x 2 , x 1 + x 3 , x 1 + x 4 , x 2 + x 3 , x 2 + x 4 , x 3 + x 4 } ⊂ R. Suppose for a contradiction that x 0 + C(X) ⊂ R; then and each term in brackets is an element of R. Since R is a Sidon set, this implies x 2 = x 3 or x 1 = x 4 , contradicting the fact that the x i are distinct. We have |R| ≥ √ m/8 and therefore we are done in the case t = 2.