Graph Coloring Clique bound: Find a graph G such that ω(G) < χ(G) = 4 For the life of me I can't seem to think of anything that works. I tried randomly combining triangle graphs and taking edges away from $K_4$, but every time I think I found a way to force the chromatic number to be 4, I find that there's actually a smaller number of colors that can be used, or that my graph actually contains $K_4$

Take a pentagon $C_5$. You need $3$ colors already. Add a vertex, and link it to all vertices of the pentagon, so that it cannot take any of these $3$ colors. You end up with $\chi=4$, and $\omega=3$.
