The lifting of graphs to 3-uniform hypergraphs and some applications to hypergraph Ramsey theory

Abstract

Given a simple graph $\Gamma$, we describe a “lifting” to a $3$-uniform hypergraph $\phi \left(\Gamma \right)$ that sends the complement of $\Gamma$ to the complement of $\phi \left(\Gamma \right)$. We consider the effects of this lifting on cycles, complete subhypergraphs, and complete subhypergraphs missing a single hyperedge. Our results lead to natural lower bounds for some hypergraph Ramsey numbers.