Ramsey properties of randomly perturbed hypergraphs

Abstract

We study Ramsey properties of randomly perturbed $3$-uniform hypergraphs. For~$t\geq 2$, write $\tilde K^{(3)}_t$ to denote the $3$-uniform {\it expanded} clique hypergraph obtained from the complete graph $K_t$ by expanding each of the edges of the latter with a new additional vertex. For an even integer $t\geq 4$, let~$M$ denote the asymmetric maximal density of the pair $(\tilde K^{(3)}_t,\tilde K^{(3)}_{t/2})$. We prove that adding a set~$F$ of random hyperedges satisfying $|F|\gg n^{3-1/M}$ to a given $n$-vertex $3$-uniform hypergraph~$H$ with non-vanishing edge density asymptotically almost surely results in a perturbed hypergraph enjoying the Ramsey property for $\tilde K^{(3)}_t$ and two colours. We conjecture that this result is asymptotically best possible with respect to the size of $F$ whenever $t\geq 6$ is even. The key tools of our proof are a new variant of the hypergraph regularity lemma accompanied with a \emph{tuple lemma} providing appropriate control over joint link graphs. Our variant combines the so called strong and the weak hypergraph regularity lemmata.