We give a short proof of a stronger form of the Johansson-Molloy theorem which relies only on the first moment method. The proof adapts a clever counting argument developed by Rosenfeld in the context of non-repetitive colourings. We then extend that result to graphs where each neighbourhood has bounded density, which improves a recent result from Davies et al. Focusing on tightening the number of colours, we obtain the best known upper bound for the chromatic number of triangle-free graphs of maximum degree $\Delta \ge 224$.

翻译：我们简短地证明只依赖第一时刻方法的约翰松-莫洛伊定理的更强的形式。 证据调整了罗森菲尔德在非重复颜色背景下开发的智能计数参数。 然后我们将这一结果扩展至每个邻里有连接密度的图形, 从而改善了戴维斯等人最近的结果。 专注于收紧颜色数量, 我们获得了最著名的无三角图色数上限为$\ Delta\ge 224美元的顶层图。