Doubly exponential growth of 3-uniform Ramsey numbers

Show that the 2-colour Ramsey number of triples, R_3(n), grows doubly exponentially in n; the best lower bound is only 2^{cn²}.

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State the problem, from Erdős Problems

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	1	+# Doubly exponential growth of 3-uniform Ramsey numbers
	2	+
	3	+Colour every triple of an $m$-element set red or blue: how large must $m$ be to guarantee $n$ points whose triples all have the same colour? The answer lies between $2^{cn^2}$ and $2^{2^n}$, an enormous gap. Erdős, Hajnal and Rado conjectured that the truth is doubly exponential, and Erdős offered \$500 for a proof.
	4	+
	5	+## Problem
	6	+
	7	+Let $R_3(n)$ be the least $m$ such that in every 2-colouring of the triples of an $m$-element set there are $n$ elements all of whose $\binom n3$ triples have the same colour. Is there a constant $c>0$ such that
	8	+
	9	+$$
	10	+R_3(n)\ge 2^{2^{cn}}\quad\text{for all large } n\,?
	11	+$$
	12	+
	13	+## Background
	14	+
	15	+For graphs, the Ramsey number $R(n)$ grows exponentially in $n$. For triples, Erdős, Hajnal and Rado (1965) proved
	16	+
	17	+$$
	18	+2^{cn^2}<R_3(n)<2^{2^{n}}
	19	+$$
	20	+
	21	+for a constant $c>0$. The lower bound comes from a random colouring, and no better construction is known for two colours. The question is a special case of the general problem of how fast hypergraph Ramsey numbers grow, and it is the key case: the Erdős–Hajnal stepping-up lemma turns lower bounds for $3$-uniform hypergraphs into lower bounds one exponential higher for each further uniformity. A doubly exponential lower bound for $R_3(n)$ would therefore give lower bounds for every uniformity $k\ge3$ that match the known upper bounds, a tower of height $k$, up to the constant in the top exponent. Erdős listed it among the combinatorial problems he most wanted to see solved.
	22	+
	23	+## What is known
	24	+
	25	+- Two colours. The bounds $2^{cn^2}<R_3(n)<2^{2^n}$ of Erdős, Hajnal and Rado (1965) remain the best known.
	26	+- Four colours. Erdős, Hajnal, Máté and Rado (1984) proved a doubly exponential lower bound when the triples are coloured with four colours.
	27	+- Three colours. Conlon, Fox and Sudakov (2010) improved an old result of Erdős and Hajnal, showing that the 3-colour Ramsey number of triples is at least $2^{n^{c\log n}}$.
	28	+- Off-diagonal numbers. Conlon, Fox and Sudakov also proved the first superexponential lower bound for $r_3(s,n)$ with $s\ge4$ fixed, answering a question of Erdős and Hajnal from 1972.
	29	+
	30	+## What would count as progress
	31	+
	32	+- A 2-colouring of the triples of a set of size $2^{2^{cn}}$ (or even $2^{n^{2+\delta}}$) with no monochromatic set of size $n$, given as an explicit or randomised construction with a proof; small cases can be checked by computer.
	33	+- A proof that $R_3(n)\le 2^{2^{o(n)}}$, which would disprove the conjecture.
	34	+- A doubly exponential lower bound for three colours.
	35	+- A formal proof against the Lean statement in [formal-conjectures](https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/564.lean).
	36	+
	37	+## References
	38	+
	39	+1. P. Erdős, A. Hajnal and R. Rado, Partition relations for cardinal numbers, Acta Math. Acad. Sci. Hungar. 16 (1965), 93–196. [doi:10.1007/BF01886396](https://doi.org/10.1007/BF01886396)
	40	+2. P. Erdős, On the combinatorial problems which I would most like to see solved, Combinatorica 1 (1981), 25–42. [doi:10.1007/BF02579174](https://doi.org/10.1007/BF02579174)
	41	+3. D. Conlon, J. Fox and B. Sudakov, Hypergraph Ramsey numbers, J. Amer. Math. Soc. 23 (2010), 247–266. [doi:10.1090/S0894-0347-09-00645-6](https://doi.org/10.1090/S0894-0347-09-00645-6)
	42	+
	43	+## Source
	44	+
	45	+Imported from [Erdős Problems 564](https://www.erdosproblems.com/564), curated by Thomas Bloom. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.