Baker, Harman, and Pintz showed that a weak form of the Prime Number Theorem holds in intervals of the form [x − x 0.525 , x] for large x. In this paper, we extend a result of Maynard and Tao concerning small gaps between primes to intervals of this length. More precisely, we prove that for any δ ∈ [0.525, 1] there exist positive integers k, d such that for sufficiently large x, the interval(log x) k pairs of consecutive primes differing by at most d. This confirms a speculation of Maynard that results on small gaps between primes can be refined to the setting of short intervals of this length.

A sunflower with r petals is a collection of r sets so that the intersection of each pair is equal to the intersection of all. Erdős and Rado proved the sunflower lemma: for any fixed r, any family of sets of size w, with at least about w w sets, must contain a sunflower. The famous sunflower conjecture is that the bound on the number of sets can be improved to c w for some constant c. In this paper, we improve the bound to about (log w) w . In fact, we prove the result for a robust notion of sunflowers, for which the bound we obtain is tight up to lower order terms.

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We show that the pattern i∈S x i , i∈S x i is partition regular over the rationals, settling Hindman's conjecture over Q.

Given a fixed graph H, a real number p ∈ (0, 1) and an infinite Erdös–Rényi graph G ∼ G(∞, p), how many adjacency queries do we have to make to find a copy of H inside G with probability at least 1/2? Determining this number f(H, p) is a variant of the subgraph query problem introduced by Ferber, Krivelevich, Sudakov and Vieira. For every graph H, we improve the trivial upper bound of f(H, p) = O(p−d), where d is the degeneracy of H, by exhibiting an algorithm that finds a copy of H in time O(p−d) as p goes to 0. Furthermore, we prove that there are 2-degenerate graphs which require p−2+o(1) queries, showing for the first time that there exist graphs H for which f(H, p) does not grow like a constant power of p−1 as p goes to 0. Finally, we answer a question of Feige, Gamarnik, Neeman, Rácz and Tetali by showing that for any δ < 2, there exists α < 2 such that one cannot find a clique of order α log2n in G(n, 1/2) in n δ queries.