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 ′ are Ramsey-equivalent if every graph G is Ramsey for H if and only if it is Ramsey for H ′. In this paper, we study the problem of determining which graphs are Ramsey-equivalent to the complete graph Kk. A famous theorem of Nešetřil and Rödl implies that any graph H which is Ramsey-equivalent to Kk must contain Kk. We prove that the only connected graph which is Ramsey-equivalent to Kk is itself. This gives a negative answer to the question of Szabó, Zumstein, and Zürcher on whether Kk is Ramsey-equivalent to Kk ·K2, the graph on k+ 1 vertices consisting of Kk 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) be the smallest minimum degree over all Ramsey minimal graphs for H. The study of s(H) was introduced by Burr, Erdős, and Lovász, where they show that s(Kk) = (k−1)2. We prove that s(Kk ·K2) = k − 1, and hence Kk and Kk ·K2 are not Ramsey-equivalent. We also address the question of which non-connected graphs are Ramsey-equivalent to Kk. Let f(k, t) be the maximum f such that the graph H = Kk + fKt, consisting of Kk and f disjoint copies of Kt, is Ramsey-equivalent to Kk. Szabó, Zumstein, and Zürcher gave a lower bound on f(k, t). We prove an upper bound on f(k, t) which is roughly within a factor 2 of the lower bound. ∗Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139-4307. Email: [email protected]. Research supported by a Packard Fellowship, by a Simons Fellowship, by NSF grant DMS-1069197, by a Sloan Foundation Fellowship, and by an MIT NEC Corporation Award. †Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139-4307. Email: [email protected]. Research supported by a National Physical Science Consortium Fellowship. ‡Department of Computer Science, University of Warwick, Coventry CV4 7AL, UK. This research was done when the author was affiliated with the Institute of Mathematics, Freie Universität Berlin, 14195 Berlin, Germany. Email: [email protected]. The author was supported by the Berlin Mathematical School. The author would like to thank the Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139-4307 for its hospitality where this work was partially carried out. §Institute of Mathematics, Goethe-Universität, 60325 Frankfurt am Main, Germany. This research was done when the author was affiliated with the Institute of Mathematics, Freie Universität Berlin. Email: [email protected] ¶Institute of Mathematics, Freie Universität Berlin, 14195 Berlin, Germany. Email: [email protected] 1 ar X iv :1 31 2. 02 99 v1 [ m at h. C O ] 2 D ec 2 01 3