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A new upper bound is given for the cycle-complete graph Ramsey number r(C,,,, K"), the smallest order for a graph which forces it to contain either a cycle of order m or a set of ri independent vertices . Then, another cycle-complete graph Ramsey number is studied, namely r( :C,,,, K") the smallest order for a graph which forces it to contain either a cycle of order 1 for some I satisfying 3 :1 !~ ,m or a set of n independent vertices . We obtain the exact value of r( :C,,,, K") for all m > n and an upper bound which applies when m is large in comparison with log n .

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In a seminal paper from 1983, Burr and Erdős started the systematic study of Ramsey numbers of cliques vs. large sparse graphs, raising a number of problems. In this paper we develop a new approach to such Ramsey problems using a mix of the Szemerédi regularity lemma, embedding of sparse graphs, Turán type stability, and other structural results. We give exact Ramsey numbers for various classes of graphs, solving five -all but one -of the Burr-Erdős problems.A weaker version of this conjecture was stated earlier by Burr in [8].Conjecture 1.2. For fixed q ≥ 1, p ≥ 3, all sufficiently large graphs of maximum degree at most q are p-good. Brandt [4] showed that for p = 3 and q ≥ 168, every q-regular graph of sufficiently large order and with sufficiently large expansion factor is a counterexample to Conjecture 1.2. In [27] we showed that, for p = 3, almost all 100-regular graphs are counterexamples to Conjecture 1.2, and thus to Conjecture 1.1.We shall answer in the affirmative all but one of the remaining questions raised in [7].Write C n for the cycle of order n. Burr and Erdős [7] showed that the wheel K 1 +C n is 3-good for n ≥ 5. This result motivated the following three questions ([7], p. 50). Question 1.3. Is the wheel K 1 + C n p-good for fixed p > 3 and n large?Recall that the kth power of a graph G is a graph G k with V (G k )= V (G) and uv ∈ E(G k ) if u and v can be joined in G by a path of length at most k. Question 1.4. Is K 1 +C k n a p-good graph for fixed k ≥ 2, p ≥ 3 and n large?

For n = 1, 2, . . . , let 6, = K2+ K,,. We pose the problem of determining the Ramsey numbers r (&, B,) and demonstrate that in many cases critical colorings are available from known examples of strongly regular graphs.