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We study the multicolor Ramsey numbers for paths and even cycles, R k (P n ) and R k (C n ), which are the smallest integers N such that every coloring of the complete graph K N has a monochromatic copy of P n or C n respectively. For a long time, R k (P n ) has only been known to lie between (k − 1 + o(1))n and (k + o(1))n. A recent breakthrough by Sárközy and later improvement by Davies, Jenssen and Roberts give an upper bound of (k − 1 4 + o(1))n. We improve the upper bound to (k − 1 2 + o(1))n. Our approach uses structural insights in connected graphs without a large matching. These insights may be of independent interest.

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For integers k ≥ 1 and n ≥ 2k + 1 the Kneser graph K(n, k) has as vertices all k-element subsets of [n] := {1, 2, . . . , n} and an edge between any two vertices (=sets) that are disjoint. The bipartite Kneser graph H(n, k) has as vertices all kelement and (n − k)-element subsets of [n] and an edge between any two vertices where one is a subset of the other. It has long been conjectured that all Kneser graphs and bipartite Kneser graphs except the Petersen graph K(5, 2) have a Hamilton cycle. The main contribution of this paper is proving this conjecture for bipartite Kneser graphs H(n, k). We also establish the existence of cycles that visit almost all vertices in Kneser graphs K(n, k) when n = 2k + o(k), generalizing and improving upon previous results on this problem.