Andrew Treglown

Monochromatic triangle tilings in dense graphs

Given a graph $H$, the Ramsey number $R(H)$ is the smallest positive integer $n$ such that every $2$-edge-colouring of $K_n$ yields a monochromatic copy of $H$. We write $mH$ to denote the union of $m$ vertex-disjoint copies of $H$. The members of the family $\{mH:m\ge1\}$ are also known as $H$-tilings. A well-known result of Burr, Erd\H os and Spencer states that $R(mK_3)=5m$ for every $m\ge2$. On the other hand, Moon proved that every $2$-edge-colouring of $K_{3m+2}$ yields a $K_3$-tiling consisting of $m$ monochromatic copies of $K_3$, for every $m\ge2$. Crucially, in Moon's result, distinct copies of $K_3$ might receive different colours. In this talk, we investigate the analogous questions where the complete host graph is replaced by a graph of large minimum degree and generalise both Moon's theorem and the Burr--Erd\H os--Spencer theorem to this setting.

This is joint work with J\'ozsef Balogh and Andrea Freschi.