Ramsey Numbers for Sets of Five Vertex Graphs With Fixed Number of Edges

Abstract

From time to time I like to hunt for small Ramsey numbers. The olassical Ramsey number r = r(G, H) is the smallest r, such that every 2-coloring of the edges of the complete graph K r contains a graph G with all edges of color 1, or a graph H with all edges of color 2. In [1] it was proposed to ask for the special Ramsey numbers for sets of graphs having fixed numbers of vertices and edges, that means to determine the smallest r = r m,n (s, t), such that every 2-coloring of the edges of K r contains any graph with m vertices and s edges of color 1, or any graph with n vertices and t edges of color 2. Since 1 ≤ s ≤ ( 2 m ) and 1 ≤ s ≤ ( 2 n ) an ( 2 m ) by ( 2 n ) rectangular array of Ramsey numbers has to be determined for graphs with m and n vertices.