Ramsey equivalence for asymmetric pairs of graphs

A graph F is Ramsey for a pair of graphs (G, H) if any red/blue-coloring of the edges of F yields a copy of G with all edges colored red or a copy of H with all edges colored blue. Two pairs of graphs are called Ramsey equivalent if they have the same collection of Ramsey graphs. The symmetric setting, that is, the case G = H, received considerable attention. This led to the open question whether there are connected graphs G and G\prime such that (G, G) and (G\prime, G\prime) are Ramsey equivalent. We make progress on the asymmetric version of this question and identify several nontrivial families of Ramsey equivalent pairs of connected graphs. Certain pairs of stars provide a first, albeit trivial, example of Ramsey equivalent pairs of connected graphs. Our first result characterizes all Ramsey equivalent pairs of stars. The rest of the paper focuses on pairs of the form (T, Kt), where T is a tree and Kt is a complete graph. We show that if T belongs to a certain family of trees, including all nontrivial stars, then (T, Kt) is Ramsey equivalent to a family of pairs of the form (T, H), where H is obtained from Kt by attaching disjoint smaller cliques to some of its vertices. In addition, we establish that for (T, H) to be Ramsey equivalent to (T, Kt), H must have roughly this form. On the other hand, we prove that for many other trees T, including all odd-diameter trees, (T, Kt) is not equivalent to any such pair, not even to the pair (T, Kt \cdot K2), where Kt \cdot K2 is a complete graph Kt with a single edge attached.

Subjects

graph Ramsey theory

Ramsey equivalence

DDC Class

510: Mathematics

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Ramsey equivalence for asymmetric pairs of graphs