Ramsey numbers and an approximation algorithm for the vertex cover problem

Summary We show two results. First we derive an upper bound for the special Ramsey numbers r_k(q) where r_k(q) is the largest number of nodes a graph without odd cycles of length bounded by 2k+1 and without an independent set of size q+1 can have. We prove The proof is constructive and yields an algorithm for computing an independent set of that size. Using this algorithm we secondly describe an O(¦V¦·¦E¦) time bounded approximation algorithm for the vertex cover problem, whose worst case ratio is , for all graphs with at most (2k+3)^k(2k+2) nodes (e.g. 1.8, if ¦V¦146000).