What is Ramsey-equivalent to a clique?

A graph G is Ramsey for H if every two-colouring of the edges of G contains a monochromatic copy of H. Two graphs H   and H′H′ are Ramsey-equivalent if every graph G is Ramsey for H   if and only if it is Ramsey for H′H′. In this paper, we study the problem of determining which graphs are Ramsey-equivalent to the complete graph KkKk. A famous theorem of Nešetřil and Rödl implies that any graph H   which is Ramsey-equivalent to KkKk must contain KkKk. We prove that the only connected graph which is Ramsey-equivalent to KkKk is itself. This gives a negative answer to the question of Szabó, Zumstein, and Zürcher on whether KkKk is Ramsey-equivalent to Kk⋅K2Kk⋅K2, the graph on k+1k+1 vertices consisting of KkKk with a pendent edge.In fact, we prove a stronger result. A graph G is Ramsey minimal for a graph H if it is Ramsey for H but no proper subgraph of G is Ramsey for H  . Let s(H)s(H) be the smallest minimum degree over all Ramsey minimal graphs for H  . The study of s(H)s(H) was introduced by Burr, Erdős, and Lovász, where they show that s(Kk)=(k−1)2s(Kk)=(k−1)2. We prove that s(Kk⋅K2)=k−1s(Kk⋅K2)=k−1, and hence KkKk and Kk⋅K2Kk⋅K2 are not Ramsey-equivalent.We also address the question of which non-connected graphs are Ramsey-equivalent to KkKk. Let f(k,t)f(k,t) be the maximum f   such that the graph H=Kk+fKtH=Kk+fKt, consisting of KkKk and f   disjoint copies of KtKt, is Ramsey-equivalent to KkKk. Szabó, Zumstein, and Zürcher gave a lower bound on f(k,t)f(k,t). We prove an upper bound on f(k,t)f(k,t) which is roughly within a factor 2 of the lower bound.