Summary: On weighted Ramsey numbers
Maria Axenovich and Ryan Martin
Department of Mathematics
Iowa State University
April 25, 2006
The weighted Ramsey number, wR(n, k), is the minimum q such that there is an assign-
ment of nonnegative real numbers (weights) to the edges of Kn with the total sum of the
weights equal to n
2 and there is a Red/Blue coloring of edges of the same Kn, such that in
any complete k-vertex subgraph H, of Kn, the sum of the weights on Red edges in H is at
most q and the sum of the weights on Blue edges in H is at most q.
This concept was introduced recently by Fujisawa and Ota, with the total weights on the
edges being equal to 1.
We provide new bounds on wR(n, k), for k 4 and n large enough and show that
determining wR(n, 3) is asymptotically equivalent to the problem of finding the fractional
packing number of monochromatic triangles in two-edge-colored complete graphs.