A well-known conjecture states that a random symmetric n × n matrix with entries in {−1, 1} is singular with probability Θ n 2 2 −n . In this paper we prove that the probability of this event is at most exp − Ω( √ n) , improving the best known bound of exp − Ω(n 1/4 √ log n) , which was obtained recently by Ferber and Jain. The main new ingredient is an inverse Littlewood-Offord theorem in Z n p that applies under very mild conditions, whose statement is inspired by the method of hypergraph containers.
We prove that for every , r ≥ 3, there exists c > 0 such that for p ≤ cn −1/m 2 (Kr ,C ) , with high probability there is a 2-edge-colouring of the random graph Gn,p with no monochromatic copy of Kr of the first colour and no monochromatic copy of C of the second colour. This is a progress on a conjecture of Kohayakawa and Kreuter.
For graphs G, H, we write G rb −→ H if any proper edge-coloring of G contains a rainbow copy of H, i.e., a copy where no color appears more than once. Kohayakawa, Konstadinidis and the last author proved that the threshold for G(n, p) H) . Previous results have matched the lower bound for this anti-Ramsey threshold for cycles and complete graphs with at least 5 vertices. Kohayakawa, Konstadinidis and the last author also presented an infinite family of graphs H for which the anti-Ramsey threshold is asymptotically smaller than n −1/m2 (H) . In this paper, we devise a framework that provides a richer and more complex family of such graphs that includes all the previously known examples.
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