Explicit Ramsey graphs

Construct explicit graphs on n vertices with no clique or independent set of size c·log n, matching Erdős's random graphs.

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shoal State the problem, from Erdős Problems 59b41b5 · 2周前历史

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1	# Explicit Ramsey graphs
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3	A random graph on $n$ vertices has, with high probability, no clique and no independent set larger than about $2\log_2 n$. Erdős asked for a construction, rather than a probabilistic existence proof, of graphs that are nearly as good: no homogeneous set of size $c\log n$. In theoretical computer science, explicit Ramsey graphs are closely tied to randomness extractors.
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5	## Problem
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7	Call a graph on $n$ vertices $t$-Ramsey if it contains no clique and no independent set of size $t$. Give an explicit construction of $t$-Ramsey graphs on $n$ vertices with $t\le c\log n$ for some constant $c>0$, for infinitely many $n$; equivalently, a constructive proof that the diagonal Ramsey number satisfies $R(k)>C^k$ for some $C>1$.
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9	"Explicit" is understood in the strong sense used in the modern literature: there is an algorithm that, given two vertices, decides whether they are adjacent in time polynomial in $\log n$.
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11	Erdős offered \$100 for this problem.
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13	## Background
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15	Erdős proved in 1947 that $R(k)>2^{k/2}$ by showing that a random colouring works, one of the first uses of the probabilistic method, and asked for a construction that achieves an exponential bound. Derandomising his argument by the method of conditional expectations gives a deterministic algorithm with running time $2^{O(k^2)}$, as noted on the problem's forum, but this does not yield graphs whose edges can be computed quickly. Recent progress has come from the construction of two-source dispersers and extractors.
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17	## What is known
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19	- Erdős's 1969 variant. He also asked for an explicit graph whose largest clique and independent set have size $o(n^{1/2})$; this is now known.
20	- Cohen (2016) constructed explicit graphs with no clique or independent set of size $2^{(\log\log n)^C}$ for a constant $C$.
21	- Li (2023) improved this to $(\log n)^{C}$ for a constant $C$, via two-source extractors for asymptotically optimal entropy. The remaining gap to the random bound is a polynomial in $\log n$.
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23	## What would count as progress
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25	- An explicit family of graphs with no homogeneous set of size $(\log n)^{C}$ for a smaller constant $C$ than in Li's construction, with the adjacency algorithm given as code.
26	- An explicit construction achieving $c\log n$, which would solve the problem.
27	- For moderate $n$, explicit graphs whose largest homogeneous sets can be certified by computer and compare well with random graphs, as evidence for a candidate construction.
28	- A formal proof against the Lean statement in [formal-conjectures](https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/78.lean).
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30	## References
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32	1. P. Erdős, Some remarks on the theory of graphs, Bull. Amer. Math. Soc. 53 (1947), 292–294. [doi:10.1090/S0002-9904-1947-08785-1](https://doi.org/10.1090/S0002-9904-1947-08785-1)
33	2. G. Cohen, Two-source dispersers for polylogarithmic entropy and improved Ramsey graphs, in Proc. 48th ACM Symposium on Theory of Computing (2016), 278–284. [doi:10.1145/2897518.2897530](https://doi.org/10.1145/2897518.2897530)
34	3. X. Li, Two source extractors for asymptotically optimal entropy, and (many) more, in Proc. 64th IEEE Symposium on Foundations of Computer Science (2023), 1271–1281. [doi:10.1109/FOCS57990.2023.00075](https://doi.org/10.1109/FOCS57990.2023.00075)
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36	## Source
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38	Imported from [Erdős Problems 78](https://www.erdosproblems.com/78), 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.