Ramsey Properties of Randomly Perturbed Hypergraphs

We study Ramsey properties of randomly perturbed 3-uniform hypergraphs. For t ≥ 2, write K^e_t⁽³⁾ to denote the 3-uniform expanded clique hypergraph obtained from the complete graph Kt by expanding each of the edges of the latter with a new additional vertex. For an even integer t ≥ 4, let M denote the asymmetric maximal density of the pair (K^e_t⁽³⁾, K^e_t/⁽³⁾₂). We prove that adding a set F of random hyperedges satisfying |F| ≫ 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 K^e_t⁽³⁾ and two colours. We conjecture that this result is asymptotically best possible with respect to the size of F whenever t ≥ 6 is even. The key tools of our proof are a new variant of the hypergraph regularity lemma accompanied with a tuple lemma providing appropriate control over joint link graphs. Our variant combines the so called strong and the weak hypergraph regularity lemmata.