I'm trying the prove the following:
Let $G$ be a simple graph with $m$ edges. Show that $\chi(G)\leq \frac{1}{2}+\sqrt{2m+\frac{1}{4}}.$
A very minute bit of algebraic manipulation shows that this is equivalent to proving $\chi(G)(\chi(G)-1)\leq 2m.$ From here I am a bit stuck. Could someone suggest a direction to head in?
Please no full solutions, just hints.