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Erdős #1156 · graph theory / chromatic number

Let GG be a random graph on nn vertices, in which every edge is included independently with probability 1/21/2.

Is there some constant CC such that that chromatic number χ(G)\chi(G) is, almost surely, concentrated on a…

MSC 05C · Graph theory

Inferred from the community field; finer classification has not been reviewed.

Source reports OPENAs of 2026-09-08Source & evidence ↓
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Problem statement

Let GG be a random graph on nn vertices, in which every edge is included independently with probability 1/21/2.

Is there some constant CC such that that chromatic number χ(G)\chi(G) is, almost surely, concentrated on at most CC values?

Is it true that, if ω(n)→∞\omega(n)\to \infty sufficiently slowly, then for every function f(n)f(n)

P(∣χ(G)−f(n)∣<ω(n))<1/2\mathbb{P}(\lvert\chi(G)-f(n)\rvert<\omega(n))<1/2

if nn is sufficiently large?

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Background

Bollobás [Bo88] proved that χ(G)∼n2log⁡2n\chi(G) \sim \frac{n}{2\log_2n} with high probability.

Shamir and Spencer [ShSp87] proved that, for any function ω(n)\omega(n) such that ω(n)/n→∞\omega(n)/\sqrt{n}\to \infty, there is a function f(n)f(n) such that

P(∣χ(G)−f(n)∣<ω(n))→1\mathbb{P}(\lvert\chi(G)-f(n)\rvert<\omega(n))\to 1

as n→∞n\to \infty. This is proved with ω(n)log⁡nn→∞\omega(n)\frac{\log n}{\sqrt{n}}\to \infty in Exercise 3 of Section 7.9 of Alon and Spencer [AlSp16] (a proof is also given by Scott [Sc17]).

Heckel [He21] proved that if ff and ω\omega are such that

P(∣χ(G)−f(n)∣<ω(n))→1\mathbb{P}(\lvert\chi(G)-f(n)\rvert<\omega(n))\to 1

as n→∞n\to \infty then, for any c<1/4c<1/4, there are infinitely many nn such that ω(n)>nc\omega(n)>n^c. This was improved to any c<1/2c<1/2 by Heckel and Riordan [HeRi23].

Progress & source assessment

The source labels this entry OPEN in the 2026-09-08 snapshot. Editorial progress notes are reproduced as source commentary, not independently verified results. Prize amounts and Lean badges at the source are not local awards or verification.

References
  1. [1]Thomas Bloom (ed.), Erdős Problems #1156 link

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