Cycle-Complete Ramsey Numbers

The Ramsey number r(C-l, K-n) is the smallest natural number N such that every red/blue edge colouring of a clique of order N contains a red cycle of length l or a blue clique of order n. In 1978, Erdos, Faudree, Rousseau, and Schelp conjectured that r(C-l, K-n) = (l - 1)(n - 1) + 1 for l >= n >= 3 provided (l, n) not equal (3, 3). We prove that, for some absolute constant C >= 1, we have r(C-l, K-n) = (l - 1)(n - 1) + 1 provided l >= Clog n/log log n. Up to the value of C this is tight since we also show that, for any epsilon > 0 and n > n(0)(epsilon), we have r(C-l, K-n) >> (l - 1)(n - 1) + 1 for all 3 <= l <= (1 - epsilon) log n/log log n. This proves the conjecture of Erdos, Faudree, Rousseau, and Schelp for large l, a stronger form of the conjecture due to Nikiforov, and answers (up to multiplicative constants) two further questions of Erdos, Faudree, Rousseau, and Schelp.