Graph Ramsey theory and the polynomial hierarchy

In the Ramsey theory of graphs F arrows (G,H) means that for every way of coloring the edges of F red and blue F will contain either a red G or a blue H. The problem ARROWING of deciding whether F arrows (G,H) lies in coNP^NP and it was shown to be coNP-hard by Burr in 1990. We prove that ARROWING is actually coNP^NP-complete, simultaneously settling a conjecture of Burr and providing a rare natural example of a problem complete for a higher level of the polynomial hierarchy. We also show that STRONG ARROWING, the version for induced subgraphs, is complete for coNP^NP under Turing-reductions.

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