Multicolour Ramsey numbers of paths and even cycles

Abstract We prove new upper bounds on the multicolour Ramsey numbers of paths and even cycles. It is well known that ( k − 1 ) n + o ( n ) ⩽ R k ( P n ) ⩽ R k ( C n ) ⩽ k n + o ( n ) . The upper bound was recently improved by Sarkozy who showed that R k ( C n ) ⩽ k − k 16 k 3 + 1 n + o ( n ) . Here we show R k ( C n ) ⩽ ( k − 1 4 ) n + o ( n ) , obtaining the first improvement to the coefficient of the linear term by an absolute constant.

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