Cameron and Erdős [6] asked whether the number of maximal sum-free sets in {1, . . . , n} is much smaller than the number of sum-free sets. In the same paper they gave a lower bound of 2 ⌊n/4⌋ for the number of maximal sum-free sets. Here, we prove the following: For each 1 ≤ i ≤ 4, there is a constant C i such that, given any n ≡ i mod 4, {1, . . . , n} contains (C i + o(1))2 n/4 maximal sum-free sets. Our proof makes use of container and removal lemmas of Green [11,12], a structural result of Deshouillers, Freiman, Sós and Temkin [7] and a recent bound on the number of subsets of integers with small sumset by Green and Morris [13]. We also discuss related results and open problems on the number of maximal sum-free subsets of abelian groups.
The classical Corrádi‐Hajnal theorem claims that every n‐vertex graph G with δ(G)≥2n/3 contains a triangle factor, when 3|n. In this paper we present two related results that both use the absorbing technique of Rödl, Ruciński and Szemerédi. Our main result determines the minimum degree condition necessary to guarantee a triangle factor in graphs with sublinear independence number. In particular, we show that if G is an n‐vertex graph with α(G)=o(n) and δ(G)≥(1/2+o(1))n, then G has a triangle factor and this is asymptotically best possible. Furthermore, it is shown for every r that if every linear size vertex set of a graph G spans quadratically many edges, and δ(G)≥(1/2+o(1))n, then G has a Kr‐factor for n sufficiently large. We also propose many related open problems whose solutions could show a relationship with Ramsey‐Turán theory. Additionally, we also consider a fractional variant of the Corrádi‐Hajnal Theorem, settling a conjecture of Balogh‐Kemkes‐Lee‐Young. Let t∈(0,1) and w:E(Kn)→[0,1]. We call a triangle t‐heavy if the sum of the weights on its edges is more than 3t. We prove that if 3|n and w is such that for every vertex v the sum of w(e) over edges e incident to v is at least (1+2t3+o(1))n, then there are n/3 vertex disjoint heavy triangles in G. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 669–693, 2016
Cameron and Erdős [6] raised the question of how many maximal sum-free sets there are in {1, . . . , n}, giving a lower bound of 2 ⌊n/4⌋ . In this paper we prove that there are in fact at most 2 (1/4+o(1))n maximal sum-free sets in {1, . . . , n}. Our proof makes use of container and removal lemmas of Green [8,9] as well as a result of Deshouillers, Freiman, Sós and Temkin [7] on the structure of sum-free sets.A fundamental notion in combinatorial number theory is that of a sum-free set: A set S of integers is sum-free if x + y ∈ S for every x, y ∈ S (note x and y are not necessarily distinct here). The topic of sum-free sets of integers has a long history. Indeed, in 1916 Schur [19] proved that, if n is sufficiently large, then any r-colouring of [n] := {1, . . . , n} yields a monochromatic triple x, y, z such that x + y = z.Note that both the set of odd numbers in [n] and the set {⌊n/2⌋ + 1, . . . , n} are maximal sum-free sets. (A sum-free subset of [n] is maximal if it is not properly contained in another sum-free subset of [n].) By considering all possible subsets of one of these maximal sum-free sets, we see that [n] contains at least 2 ⌈n/2⌉ sum-free sets. Cameron and Erdős [5] conjectured that in fact [n] contains only O(2 n/2 ) sum-free sets. The conjecture was proven independently by Green [8] and Sapozhenko [16]. Recently, a refinement of the Cameron-Erdős conjecture was proven in [1], giving an upper bound on the number of sum-free sets in [n] of size m (for each 1 ≤ m ≤ ⌈n/2⌉).Let f (n) denote the number of sum-free subsets of [n] and f max (n) denote the number of maximal sum-free subsets of [n]. Recall that the sum-free subsets of [n] described above lie in *
Addressing a question of Cameron and Erdős, we show that, for infinitely many values of n, the number of subsets of {1, 2, . . . , n} that do not contain a k-term arithmetic progression is at most 2 O(r k (n)) , where r k (n) is the maximum cardinality of a subset of {1, 2, . . . , n} without a k-term arithmetic progression. This bound is optimal up to a constant factor in the exponent. For all values of n, we prove a weaker bound, which is nevertheless sufficient to transfer the current best upper bound on r k (n) to the sparse random setting. To achieve these bounds, we establish a new supersaturation result, which roughly states that sets of size Θ(r k (n)) contain superlinearly many k-term arithmetic progressions.For integers r and k, Erdős asked whether there is a set of integers S with no (k+1)term arithmetic progression, but such that any r-coloring of S yields a monochromatic k-term arithmetic progression. Nešetřil and Rödl, and independently Spencer, answered this question affirmatively. We show the following density version: for every k ≥ 3 and δ > 0, there exists a reasonably dense subset of primes S with no (k+1)-term arithmetic progression, yet every U ⊆ S of size |U | ≥ δ|S| contains a k-term arithmetic progression.Our proof uses the hypergraph container method, which has proven to be a very powerful tool in extremal combinatorics. The idea behind the container method is to have a small certificate set to describe a large independent set. We give two further applications in the appendix using this idea.
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