We show that for c >= 2.4682, a random graph on n vertices with c n (1+o(1)) edges almost surely has no 3-colouring. This improves on the current best upper bound of 2.4947.
@article{arxiv.math/0209087,
title = {On the non-3-colourability of random graphs},
author = {O. Dubois and J. Mandler},
journal= {arXiv preprint arXiv:math/0209087},
year = {2007}
}