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Erdős problem 579 · Conjectures.io
Combinatorics
Erdős problem 579
Let δ>0. If n is sufficiently large and G is a graph on n vertices with no
K2,2,2 (the octahedron) and at least δn2 edges, must G contain an independent
set of size ≫δn?
This is a problem of Erdős, Hajnal, Sós, and Szemerédi [EHSS83]. It is **open**; they proved
the statement for δ>1/8 (see erdos_579.variants.ehss_large_delta), and the
difficulty is to push the edge-density threshold down to an arbitrary δ>0.
Here K2,2,2 is the complete tripartite graph with all parts of size 2, encoded as
completeMultipartiteGraph (fun _ : Fin 3 => Fin 2); "contains no K2,2,2" is expressed
via SimpleGraph.Free.
0 α bounty·nobody has started
week of 11 Oct 2025 · nothing publishedweek of 18 Oct 2025 · nothing publishedweek of 25 Oct 2025 · nothing publishedweek of 1 Nov 2025 · nothing publishedweek of 8 Nov 2025 · nothing publishedweek of 15 Nov 2025 · nothing published
Solve it
Nobody has attempted this. A complete proof or refutation takes the whole bounty.
11 Oct 2025nothing published against this one yet3 Oct 2026
erdosproblems.com/579
[EHSS83] P. Erdős, A. Hajnal, V. T. Sós and E. Szemerédi, More results on Ramsey–Turán type problems, Combinatorica 3 (1983), 69–81.
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