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OP-9349OpenGraph theory

Erdős #1104 · graph theory / chromatic number

Let f(n)f(n) be the maximum possible chromatic number of a triangle-free graph on nn vertices. Estimate f(n)f(n).

MSC 05C · Graph theory

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

Let f(n)f(n) be the maximum possible chromatic number of a triangle-free graph on nn vertices. Estimate f(n)f(n).

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Background

The best bounds available are

(1−o(1))(n/log⁡n)1/2≤f(n)≤(2+o(1))(n/log⁡n)1/2.(1-o(1))(n/\log n)^{1/2}\leq f(n) \leq (2+o(1))(n/\log n)^{1/2}.

The upper bound is due to Davies and Illingworth [DaIl22], the lower bound follows from a construction of Hefty, Horn, King, and Pfender [HHKP25].

One can ask a similar question for the maximum possible chromatic number of a triangle-free graph on mm edges. Let this be g(m)g(m). Davies and Illingworth [DaIl22] prove

g(m)≤(35/3+o(1))(m(log⁡m)2)1/3.g(m) \leq (3^{5/3}+o(1))\left(\frac{m}{(\log m)^2}\right)^{1/3}.

Kim [Ki95] gave a construction which implies g(m)≫(m/(log⁡m)2)1/3g(m) \gg (m/(\log m)^2)^{1/3}.

The function f(n)f(n) is the inverse to the function h3(k)h_3(k) considered in [1013].

A generalisation of f(n)f(n) is considered in [920].

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References
  1. [1]Thomas Bloom (ed.), Erdős Problems #1104 link
  2. [2]Upstream Lean 4 statement for Erdős #1104 linkExternal statement reference. Not compiled or verified by this platform; upstream files may contain open proof obligations.

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