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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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Hajós conjectured in 1968 that every Eulerian graph can be decomposed into at most ⌊(n − 1)/2⌋ edge-disjoint cycles. This has been confirmed for some special graph classes, but the general case remains open. In a sequence of papers by Bienia and Meyniel (1986), Dean (1986), and Bollobás and Scott (1996 it was analogously conjectured that every directed Eulerian graph can be decomposed into O(n) cycles.In this paper, we show that every directed Eulerian graph can be decomposed into at most O(n log ∆) disjoint cycles, thus making progress towards the conjecture by Bollobás and Scott. Our approach is based on finding heavy cycles in certain edge-weightings of directed graphs. As a further consequence of our techniques, we prove that for every edge-weighted digraph in which every vertex has out-weight at least 1, there exists a cycle with weight at least Ω(log log n/log n), thus resolving a question by Bollobás and Scott.