Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)

Date and Time: Thursday, March 27, 2014, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Anita Liebenau (University of Warwick)

What is Ramsey-equivalent to the 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'. We study the problem of determining which graphs are Ramsey-equivalent to the complete graph K_k. A famous theorem of Folkman implies that any graph H which is Ramsey-equivalent to K_k must contain K_k.

In this talk, we show that the only connected graph which is Ramsey-equivalent to K_k is itself. This gives a negative answer to the question of Szabó, Zumstein, and Zürcher on whether K_k is Ramsey-equivalent to K_k.K₂, the graph on k+1 vertices consisting of K_k with a pendant edge. 

We also address the question of which non-connected graphs are Ramsey-equivalent to K_k. Let f(k,t) be the maximum f such that the graph H=K_k+fK_t, consisting of K_k and f disjoint copies of K_t, is Ramsey-equivalent to K_k. 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. 

Joint work with J. Fox, A. Grinshpun, Y. Person and T. Szabó.

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