Stability results for random discrete structures

Abstract

Two years ago, Conlon and Gowers, and Schacht proved general theorems that allow one to transfer a large class of extremal combinatorial results from the deterministic to the probabilistic setting. Even though the two papers solve the same set of long-standing open problems in probabilistic combinatorics, the methods used in them vary significantly and therefore yield results that are not comparable in certain aspects. the theorem of Schacht can be applied in a more general setting and yields stronger probability estimates, whereas the one of Conlon and Gowers also implies random versions of some structural statements such as the famous stability theorem of Erdős and Simonovits. In this paper, we bridge the gap between these two transference theorems. Building on the approach of Schacht, we prove a general theorem that allows one to transfer deterministic stability results to the probabilistic setting that is somewhat more general and stronger than the one obtained by Conlon and Gowers. We then use this theorem to derive several new results, among them a random version of the Erdős-Simonovits stability theorem for arbitrary graphs. the main new idea, a refined approach to multiple exposure when considering subsets of binomial random sets, may be of independent interest.

Independent sets in hypergraphs

“…Moreover, it was proved by Luczak [45] that Theorem 1.8 implies Theorems 1.5 and 1.6. The work of Conlon and Gowers [12] and Schacht [58] (see also [55]), as well as this work, have shown that one does not need to appeal to the sparse regularity lemma and to the K LR conjecture in order to prove such extremal statements in random graphs. Nevertheless, there is still a plentitude of beautiful corollaries of the conjecture that cannot (yet) be proved by other means.…”

“…We then, in Section 7, use it to derive a natural generalization of Theorem 1.3 to t-balanced t-uniform hypergraphs, Theorem 7.2, which was also first proved in [12] and [58]. Our methods also yield the following sparse random analogue of the famous stability theorem of Erdős and Simonovits [14,60], originally proved by Conlon and Gowers [12] in the case when H is strictly 2-balanced and then extended to arbitrary H by Samotij [55], who adapted the argument of Schacht [58] for this purpose. Theorem 1.4.…”

“…In Section 6, we prove 'random' and 'counting' versions of the general stability result of Conlon and Gowers [12] in a form that is easily comparable with [55,Theorem 3.4]. In Section 7, we discuss several applications of Theorem 2.2 in the context of the Turán problem in sparse random graphs.…”

Extremal Results in Random Graphs

“…It was conjectured by Kohayakawa, Luczak, and Rödl [65, Conjecture 1(ii )] that such a statement is true as long as p is of the order of magnitude given in the 1-statement of the threshold in Theorem 2.2. Conlon and Gowers [22] verified this conjecture for strictly 2-balanced graphs F , and Samotij [100] adapted and simplified the approach of Schacht [102] to obtain such a result for all graphs F . This led to the following probabilistic version of the Erdős-Simonovits stability theorem.…”

“…We restricted the discussion to extremal question in random graphs. However, the results of Conlon and Gowers [22] and Schacht [102] and also the subsequent work of Samotij [100], Balogh, Morris, Samotij [8], and Saxton and Thomason [101] applied in a more general context and led to extremal results for random hypergraphs and random subsets of the integers. Here we state a probabilistic version of Szemerédi's theorem on arithemtic progressions [110] (see Theorem 5.1 below).…”