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Simonovits-Sós Conjecture

A family F subset= 2^([n]) is 3AP-intersecting if the intersection of every two members contains a nonconstant three-term arithmetic progression. The Simonovits-Sós conjecture (Chung et al. 1986) asserts that

|F|<=2^(n-3).

Equality is attained by the family of all subsets containing one fixed three-term arithmetic progression.

Keevash (2026) obtained the first bound separated from the trivial 2^(n-1) estimate, proving that there is an absolute constant c>0 such that

|F|<=(1/2-c)2^n.

The conjectured sharp bound remains open. Keevash (2026) reports that GPT-6 Astra found the proof after he supplied a hint, and that he then checked and rewrote it. As of Sep. 22, 2026, independent specialist review had not been reported.

See also

Arithmetic Progression

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References

Chung, F. R. K.; Graham, R. L.; Frankl, P.; and Shearer, J. B. "Some Intersection Theorems for Ordered Sets and Graphs." J. Combin. Theory Ser. A 43, 23-37, 1986.Keevash, P. "A Non-Trivial Bound for 3AP-Intersecting Families." 16 Sep 2026. https://arxiv.org/abs/2609.18870.

Cite this as:

Weisstein, Eric W. "Simonovits-Sós Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Simonovits-SosConjecture.html

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