Ramsey-type numbers involving graphs and hypergraphs with large girth

For a set of integers S, define ( S APk ) to be the k-uniform hypergraph with vertex set S and hyperedges corresponding to the set of all arithmetic progression of length k in S. Similarly, for a graph H, define ( H Kk ) to be the ( k 2 ) -uniform hypergraph on the vertex set E(H) with hyperedges corresponding to the edge sets of all copies of Kk in H. Also, we say that a k-uniform hypergraph has girth at least g if any subset of h edges (2 ≤ h < g) span at least (k − 1)h + 1 vertices. For all integers k and `, we establish the existence of a relatively small graph H having girth k and the property that every `-coloring of the edges of H yields a monochromatic copy of Ck. We also show that for all integers k, `, and g, there exists a relatively small set S ⊂ N such that the related hypergraph ( S APk ) has girth g and each `-coloring of S yields a monochromatic arithmetic progression of length k. Finally, for all integers k, `, and g, we establish the existence of a relatively small graph H such that the associated hypergraph ( H Kk ) has girth g and each `-coloring of the edges of H yields a monochromatic copy of Kk. Our proofs give improved (and the first explicit) numerical bounds on the size of these objects.