The induced Ramsey number R_ind(H) of a graph H is the minimum number N such that there exists a graph with N vertices for which all red/blue colorings of its edges contain a monochromatic induced copy of H. In this talk I'll show there exists an absolute constant C > 0 such that, for every graph H on k vertices, these numbers satisfy R_ind(H) ≤ 2^Ck^. This resolves a conjecture of Erdős from 1975.

This is joint work with Lucas Aragão, Gabriel Dahia, Rafael Filipe, João Marciano.