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OP-9305OpenGraph theory

Erdős #1029 · graph theory / ramsey theory

If R(k)R(k) is the Ramsey number for KkK_k, the minimal nn such that every 22-colouring of the edges of KnK_n contains a monochromatic copy of KkK_k, then

R(k)k2k/2→∞.\frac{R(k)}{k2^{k/2}}\to \infty.
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

If R(k)R(k) is the Ramsey number for KkK_k, the minimal nn such that every 22-colouring of the edges of KnK_n contains a monochromatic copy of KkK_k, then

R(k)k2k/2→∞.\frac{R(k)}{k2^{k/2}}\to \infty.

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Background

In [Er93] Erdős offers $100 for a proof of this and $1000 for a disproof, but says 'this last offer is to some extent phoney: I am sure that [this] is true (but I have been wrong before).'

Erdős and Szekeres [ErSz35] proved

k2k/2≪R(k)≤(2k−1k−1).k2^{k/2} \ll R(k) \leq \binom{2k-1}{k-1}.

One of the first applications of the probabilistic method pioneered by Erdős gives

R(k)≥(1+o(1))12ek2k/2,R(k) \geq (1+o(1))\frac{1}{\sqrt{2}e}k2^{k/2},

which Spencer [Sp75] improved by a factor of 22 to

R(k)≥(1+o(1))2ek2k/2.R(k) \geq (1+o(1))\frac{\sqrt{2}}{e}k2^{k/2}.

See also [77] for a more general problem concerning lim⁡R(k)1/k\lim R(k)^{1/k}, and discussion of upper bounds for R(k)R(k).

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 #1029 link
  2. [2]Formal Conjectures — Erdős Problem 1029 link

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