Shamir and Spencer proved in the 1980s that the chromatic number of the binomial random graph G(n,p) is concentrated in an interval of length at most \omega\sqrt{n}, and in the 1990s Alon showed that an interval of length \omega\sqrt{n}/\log n suffices for constant edge-probabilities p \in (0,1). We prove a similar logarithmic improvement of the Shamir-Spencer concentration results for the sparse case p=p(n) \to 0, and uncover a surprising concentration `jump' of the chromatic number in the very dense case p=p(n) \to 1.

翻译：Shamir 和 Spencer 在20世纪80年代证明了二项随机图 G(n,p) 的色数集中于长度至多为 ω√n 的区间内，而 Alon 在90年代指出，对于常值边概率 p∈(0,1)，长度为 ω√n/log n 的区间即足够。针对稀疏情形 p=p(n)→0，我们证明了 Shamir-Spencer 集中性结果的类似对数改进，并在极稠密情形 p=p(n)→1 下揭示了色数集中性的一个令人惊讶的“跃变”。