The Matching Kneser Graph Conjecture For High Chromatic Numbers

Abstract

\noindent In this paper, we show that for any positive integers $r$, $k$, $\Theta$, and $\Gamma$ such that $k\ge 2$ and $r\ge k+\Gamma$, there exists a connected graph $G$ for which $$\begin{array}{llcr}\omega (G)=\chi (G)=k,& \chi \left(G,r{K}_2\right)=\Theta ,& \mathrm{and}& |E(G)|-\mathrm{ex}\left(G,r{K}_2\right)=\Theta +\Gamma .\end{array}$$