Combinatorial theorems in sparse random sets

Abstract

We describe recent advances in the study of random analogues of combinatorial theorems.

Sharp thresholds for Ramsey properties of strictly balanced nearly bipartite graphs

“…More recently, Friedgut, Hàn, Person and Schacht [8] studied van der Waerden's property in random subsets of Z/nZ and established a sharp threshold for this property. Essentially the same proof yields the sharpness of the threshold for the Ramsey properties of strictly balanced (see (2) below) k-partite k-uniform hypergraphs and, hence, in particular for even cycles in graphs and two colors.…”

“…In fact, the cores then correspond to the complements of the almost independent sets from J given by the Saxton-Thomason theorem (Theorem 6). This yields a small family C of subsets of V (H), that means of size 2 o(pn 2 ) , such that the elements C ∈ C are not too small and every hitting set of H contains at least one element from C.…”

“…Since for every k ≥ 3 the cycle C k of length k is strictly balanced and nearly bipartite, the following result includes Theorem 1 as a special case. We remark that here we defined d 2 (K 2 ) = 1 in (2). As a consequence it follows that…”

“…We know by the condition |e 1 ∩ e 2 | = 1 that X ≤ 2p 2 n 3 , thus we get with X instead of X in (22) a factor of 2p 2 n 3 instead of pn 2 …”

Transference for the Erdős–ko–rado Theorem

“…Some of the early highlights include a version of Mantel's theorem for random graphs proved by Babai, Simonovits and Spencer [1], the Ramsey theoretic results of Rödl and Ruciński [25,26], and a random analogue of Szemerédi's theorem due to Kohayakawa, Łuczak and Rödl [20]. Very general transference theorems have since been proved by Conlon and Gowers [8], Schacht [29], Balogh, Morris and Samotij [4], and Saxton and Thomason [28]. The surveys of Łuczak [24] and Rödl and Schacht [27] provide a detailed account of such results.…”