No abstract

In 1987, Kolaitis, Prömel and Rothschild proved that, for every fixed r ∈ N, almost every n-vertex K r+1 -free graph is r-partite. In this paper we extend this result to all functions r = r(n) with r (log n) 1/4 . The proof combines a new (close to sharp) supersaturation version of the Erdős-Simonovits stability theorem, the hypergraph container method, and a counting technique developed by Balogh, Bollobás and Simonovits.

We count orientations of G(n, p) avoiding certain classes of oriented graphs. In particular, we study T r (n, p), the number of orientations of the binomial random graph G(n, p) in which every copy of K r is transitive, and S r (n, p), the number of orientations of G(n, p) containing no strongly connected copy of K r . We give the correct order of growth of log T r (n, p) and log S r (n, p) up to polylogarithmic factors; for orientations with no cyclic triangle, this significantly improves a result of Allen, Kohayakawa, Mota, and Parente. We also discuss the problem for a single forbidden oriented graph, and state a number of open problems and conjectures. KEYWORDS restricted orientations; random graphs; forbidden digraphs 1 Random Struct Alg. 2020;56:1016-1030.wileyonlinelibrary.com/journal/rsa

In this paper we determine the number and typical structure of sets of integers with bounded doubling. In particular, improving recent results of Green and Morris, and of Mazur, we show that the following holds for every fixed λ > 2 and every karbitrarily slowly), then almost all sets A ⊂ [n] with |A| = k and |A + A| λk are contained in an arithmetic progression of length λk/2 + ω. The first author was partially supported by CNPq, the second author by PRPq/UFMG (ADRC 11/2017), the third author by CNPq (Proc. 303275/2013-8) and FAPERJ (Proc. 201.598/2014), the fourth author by a CNPq bolsa PDJ, and the fifth author by CAPES.1 That is, a set of the form P = a + i 1 d 1 + • • • + i s d s : i j ∈ {0, . . . , k j } for some a, d 1 , .