m-subsets of[n]` that contain no homothetic copy ofF.
Finally, we mention one more straightforward application of Theorem 44, which is a sparse version of a result of S´ark ¨ozy [115] and F ¨urstenberg [66] on square differences. A robust version of it was proved by Hamel and Łaba [78, Theorem 3.1] using a Varnavides-type averaging argument. The following theo-rem improves Theotheo-rem 1.2 of [78].
Theorem 58.For every positiveβ, there exist constantsCandn0such that the following holds. For everyn ∈Nwithn >n0, ifm>C√
n, then there are at most βn
m
m-subsets of[n]that contain no pair{x, y}such thatx−yis a perfect square.
2.4 Extremal results for sparse sets
In this subsection, we shall deduce from Theorem 44 two versions of the general transference theorem of Schacht [117, Theorem 3.3]. We remind the reader that a
statement very similar to Schacht’s theorem was proved independently by Con-lon and Gowers [43]. For the benefit of the readers who are familiar with [117], we shall state it using the terminology used there.
Definition 59. LetH = (Hn)n∈Nbe a sequence ofk-uniform hypergraphs and let α ∈ [0,1). We say that H is α-dense if the following is true: For every positive δ, there exist positive ε and n0 such that for every n with n > n0 and every U ⊆V(Hn)with|U|>(α+δ)v(Hn), we have
e(Hn[U])>εe(Hn).
Let us remark here that Definition 43 is a generalization of Definition 59. In-deed, ifFδdenotes the collection of all subsets ofV(Hn)with at least(α+δ)v(Hn) elements, then a sequenceH of hypergraphs isα-dense if and only if for every positiveδ, there exists a positiveεsuch that for all sufficiently largen, the hyper-graphHnis(Fδ, ε)-dense.
We start with the ‘random’ version of our extremal result, which is a slight weakening of Theorem 3.3 of [117], see the discussion below.
Theorem 60. LetHbe a sequence ofk-uniform hypergraphs, letα∈[0,1), and letcand c0 be positive constants. Suppose thatp ∈[0,1]Nis a sequence of probabilities such that for all sufficiently largen ∈N, we havepk−1n e(Hn)>c0v(Hn)and for every` ∈[k−1],
∆`(Hn)6c·min
p`−kn , p`−1n e(Hn) v(Hn)
. (2.19)
IfHisα-dense, then the following holds. For every positiveδ, there exists a constantC such that ifqn>Cpnandqnv(Hn)→ ∞asn→ ∞, then a.a.s.
α Hn[V(Hn)qn]
6(α+δ)qnv(Hn).
We note that the probability bounds implicit in the ‘asymptotically almost surely’ statement that we obtain are, as in [117], optimal, that is, they decay ex-ponentially inpnv(Hn).
Remark 61. We remark that the only difference between Theorem 60 and The-orem 3.3 of [117] are the assumptions on the hypergraph sequence H, which
are somewhat more restrictive here. In fact, it turns out that the conditions pk−1n e(Hn) > c0v(Hn) and ∆`(Hn) 6 c·p`−1n e(Hv(Hn)
n) for every` ∈ [k−1]are essen-tially equivalent to the condition thatHis(K,p)-bounded (see below), whereas the condition ∆`(Hn) 6 cp`−kn for every ` ∈ [k − 1], which is essential in the theorem above, is not needed in [117].
To be more precise, let us first recall from [117] that a sequence ofk-uniform hypergraphsHis said to be(K,p)-bounded if
µi(Hn, q) = E
Conversely, suppose thatHis(K,p)-bounded and observe that µk−1(Hn, pn)>pk−1n e(Hn).
Thus, settingc0 = 1/K, it follows thatpk−1n e(Hn) >c0v(Hn). Moreover, we claim that for every positiveε, there is a constantcthat depends only onK, k, and ε, such that for all sufficiently largen, the hypergraphHncontains a subhypergraph H0n⊆ Hnsatisfyinge(H0n)>(1−ε)e(Hn)and
∆`(Hn0)6c·p`−1n e(Hn0)
v(H0n) for every `∈[k−1]. (2.20)
Indeed, fix a large constant c and suppose that Hn does not contain a subhy-pergraph with at least (1−ε)e(Hn) edges that satisfies (2.20). In this case, let us greedily construct a hypergraphH00n as follows. Start withH0n = Hn and H00n empty. Whenever there is an`-setT ⊆V(Hn), for some`∈[k−1], whose degree inH0nexceeds the right-hand side of (2.20), then move an arbitrary edge contain-ingT fromH0ntoH00n. By our assumption, when the process terminates,H00n will contain more thanεe(Hn)edges. Now, ifcis sufficiently large (as a function ofε, k, andK), then for somei∈[k−1]we have
µi(Hn, pn)> e(H00n)
k−1 ·c·p`−1n e(H0n)
v(Hn0)·p2i−(`−1)n > Kp2in e(Hn)2 v(Hn), which is a contradiction.
Finally, note that, trivially, ifHnis(F,2ε)-dense for some familyF ⊆ P(V(Hn)), then everyH0nwithe(H0n)>(1−ε)e(Hn)is(F, ε)-dense. It follows from the above discussion that, up to the value of the involved constants, the only difference be-tween Theorem 60 and Theorem 3.3 of [117] is in the additional assumption that
∆`(Hn)6cp`−kn for every`∈[k−1].
Our methods also yield the following ‘counting’ analogue of Theorem 60, a generalization of Theorem 32 that does not follow from the methods of Schacht [117] or Conlon and Gowers [43] and, in the caseα = 0, can be thought of as a strengthening of Theorem 60, see Corollary 33.
Theorem 62. LetHbe a sequence ofk-uniform hypergraphs, letα∈[0,1), and letcand c0 be positive constants. Suppose thatp ∈[0,1]Nis a sequence of probabilities such that for all sufficiently largen ∈N, we havepk−1n e(Hn)>c0v(Hn)and for every` ∈[k−1],
∆`(Hn)6c·min
p`−kn , p`−1n e(Hn) v(Hn)
.
IfHisα-dense, then the following holds. For every positiveδ, there exists a constantC such that for all sufficiently largen, ifm >Cpnv(Hn), then
|I(Hn, m)|6
(α+δ)v(Hn) m
.
Proof of Theorem 60. Letα∈[0,1), letk∈N, letp∈[0,1]N, and letHbe a sequence ofk-uniform hypergraphs as in the statement of Theorem 60. Furthermore, sup-pose thatHisα-dense and fix some positiveδ; without loss of generality, we may assume thatδ is sufficiently small. Let n ∈ N be sufficiently large, letδ0 = δ/3, and let F denote the family of all subsets of V(Hn) with at least (α +δ0)v(Hn) elements. SinceHnis α-dense, it follows thatHnis(F, ε)-dense for some small positive ε that does not depend on n. Let C0 = C44(k, ε, c, c0). By Theorem 44, in the right-hand side of (2.21) as follows:
Pr
Hence, by Chernoff’s inequality, we have Pr
Finally, note that since|S|6δ3qv(Hn)for everyS ∈ S, and using (2.14),
Putting (2.21), (2.22), (2.23), and (2.24) together, we obtain Pr se-quence of k-uniform hypergraphs as in the statement of Theorem 60. Further-more, suppose that H is α-dense and fix some positive δ. Let n be sufficiently large, letδ0 =δ/2, and letF denote the family of all subsets ofV(Hn)with at least toF and hence there is no independent set of sizem. Therefore, from now on we may assume that m < (α+δ0)v(Hn) = (α+δ/2)v(Hn). It follows, using (2.15)
Settings =|S|and recalling thats6δ2mby (2.25), we obtain
provided thatδis sufficiently small. It follows that
|I(Hn, m)|=X