Generalised Ramsey numbers and Bruhat order on involutions

Ramsey theory is a classic topic in graph theory, and Paper A deals with two of its most fundamental problems: to compute Ramsey numbers and to characterise critical graphs. More precisely, we study generalised Ramsey numbers for two sets ?1 and ?2 of cycles. We determine, in particular, all generalised Ramsey numbers R(?1, ?2) such that ?1 or ?2 contains a cycle of length at most 6, or the shortest cycle in each set is even. This generalises previous results of Erdös, Faudree, Rosta, Rousseau, and Schelp. Furthermore, we give a conjecture for the general case. We also characterise many (?1, ?2)-critical graphs. As special cases, we obtain complete characterisations of all (Cn,C3)-critical graphs for n ? 5, and all (Cn,C5)-critical graphs for n ? 6.

In Paper B, we study the combinatorics of certain partially ordered sets. These posets are unions of conjugacy classes of involutions in the symmetric group Sn, with the order induced by the Bruhat order on Sn. We obtain a complete characterisation of the posets that are graded. In particular, we prove that the set of involutions with exactly one fixed point is graded, which settles a conjecture of Hultman in the affirmative. When the posets are graded, we give their rank functions. We also give a short, new proof of the EL-shellability of the set of fixed-point-free involutions, recently proved by Can, Cherniavsky, and Twelbeck.