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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 , .

The set-colouring Ramsey number $R_{r,s}(k)$ is defined to be the minimum $n$ such that if each edge of the complete graph $K_n$ is assigned a set of $s$ colours from $\{1,\ldots,r\}$, then one of the colours contains a monochromatic clique of size $k$. The case $s = 1$ is the usual $r$-colour Ramsey number, and the case $s = r - 1$ was studied by Erd\H{o}s, Hajnal and Rado in 1965, and by Erd\H{o}s and Szemerédi in 1972. The first significant results for general $s$ were obtained only recently, by Conlon, Fox, He, Mubayi, Suk and Verstra\"ete, who showed that $R_{r,s}(k) = 2^{\Theta(kr)}$ if $s/r$ is bounded away from $0$ and $1$. In the range $s = r - o(r)$, however, their upper and lower bounds diverge significantly. In this note we introduce a new (random) colouring, and use it to determine $R_{r,s}(k)$ up to polylogarithmic factors in the exponent for essentially all $r$, $s$ and $k$.

The set‐coloring Ramsey number is defined to be the minimum such that if each edge of the complete graph is assigned a set of colors from , then one of the colors contains a monochromatic clique of size . The case is the usual ‐color Ramsey number, and the case was studied by Erdős, Hajnal and Rado in 1965, and by Erdős and Szemerédi in 1972. The first significant results for general were obtained only recently, by Conlon, Fox, He, Mubayi, Suk and Verstraëte, who showed that if is bounded away from 0 and 1. In the range , however, their upper and lower bounds diverge significantly. In this note we introduce a new (random) coloring, and use it to determine up to polylogarithmic factors in the exponent for essentially all , , and .

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