On weighted Ramsey numbers

TL;DR

This paper investigates the weighted Ramsey number, providing new bounds for larger k and linking the case k=3 to the fractional packing number of monochromatic triangles, advancing understanding of weighted edge colorings.

Contribution

It introduces improved bounds for weighted Ramsey numbers for k≥4 and connects the k=3 case to fractional packing problems, offering new insights into weighted graph colorings.

Findings

New bounds established for wR(n,k) when k≥4 and n large

Asymptotic equivalence shown between wR(n,3) and fractional packing of monochromatic triangles

Enhanced understanding of weighted edge colorings in complete graphs

Abstract

The weighted Ramsey number, ${\rm wR}(n,k)$, is the minimum $q$ such that there is an assignment of nonnegative real numbers (weights) to the edges of $K_n$ with the total sum of the weights equal to ${n\choose 2}$ and there is a Red/Blue coloring of edges of the same $K_n$, such that in any complete $k$-vertex subgraph $H$, of $K_n$, 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. We provide new bounds on ${\rm wR}(n,k)$, for $k\geq 4$ and $n$ large enough and show that determining ${\rm wR}(n,3)$ is asymptotically equivalent to the problem of finding the fractional packing number of monochromatic triangles in colorings of edges of complete graphs with two colors.