Graph without four cycle and large chromatic number

Question

Give an example of graph without four cycle with chromatic number greater than $3$.

Answer 1

This relates to Steinberg's conjecture. It conjectures that a planar graph wiht no $C_4$ and no $C_5$ is 3-colorable. The added $C_5$ property is there because there are planar graphs with no $C_4$ that are not 3-colorable. Here's what they give :

There are probably smaller examples if we don't restrict ourselves to planar graphs, but it doesn't seem easy to construct.