Marcelo Sales, On the Ramsey number of Daisies and other hypergraphs

Given a $k$-uniform hypergraph $H$, the Ramsey number $R(H;q)$ is the smallest integer $N$ such that any $q$-coloring of the edges of the complete $k$-uniform hypergraph on $N$ vertices contains a monochromatic copy of $H$.

When $H$ is a complete hypergraph, a classical argument of Erdős, Hajnal, and Rado reduces the general problem to the case of uniformity $k = 3$. In this talk, we will survey constructions that lift Ramsey numbers to higher uniformities and discuss recent progress on quantitative bounds for $R(H;q)$ for certain families of hypergraphs.

This is joint work with Ayush Basu, Dániel Dobák, Pavel Pudlák, and Vojtěch Rödl.