Graph Theory and Combinatorics Seminar

Title: Ramsey numbers of loose triangles versus cliques

Abstract: An $r$-uniform loose triangle $C_3$ consists of three edges that intersect pairwise in exactly one vertex and have no vertex common to all three. For graphs, this is the usual triangle, and the Ramsey number $R(C_3,K_t)$ has order of magnitude $t^2/\log t$ by celebrated results of Ajtai, Koml\'os, and Szemer\'edi and of Kim.

Kostochka, Mubayi, and Verstra\"ete proved that $R(C_3,K_t^{(r)})=O_r(t^{3/2})$ for every fixed $r\ge3$ and showed that this bound is sharp up to a logarithmic factor. Motivated by the logarithmic saving in the graph case, they conjectured in 2013 that the upper bound could be improved to $o(t^{3/2})$. This conjecture has since received significant attention and has been reiterated in several papers. We prove the conjecture by showing that, for every fixed $r\ge3$, $R(C_3,K_t^{(r)}) =O_r\left(\frac{t^{3/2}}{\sqrt{\log\log t}}\right)$.

This is joint work with Ayush Basu and Abhishek Dhawan.