Simonovits–Sós 3AP-intersecting-family conjecture

For every positive integer $n$ , let $[n]=\{1,\ldots,n\}$ and let $\mathcal{F}\subseteq 2^{[n]}$ be a family such that for every $A,B\in\mathcal{F}$ , the intersection $A\cap B$ contains a non-trivial three-term arithmetic progression; that is, there exist integers $a,d$ with $d\ne 0$ and $\{a,a+d,a+2d\}\subseteq A\cap B$ . Then $|\mathcal{F}|\le 2^{n-3}$ , and this bound is conjectured to be best possible.

References

Primary source

arXiv

Additional references

A non-trivial bound for 3AP-intersecting families — arXiv — Peter Keevash

additive-combinatorics extremal-combinatorics

Posted 2 weeks ago · edited 2 weeks ago

Progress summary

Refreshed 2 weeks ago

Claimed progress

A September 2026 paper reports the first nontrivial improvement toward the conjectured bound, but the conjecture remains unsettled.

The Simonovits–Sós conjecture predicts that a 3AP-intersecting family has maximum size $2^{n-3}$ . The available report gives progress toward this bound but does not establish the conjectured sharp constant.

September 2026 development

Peter Keevash reports a uniform gap below one half for 3AP-intersecting families and extends the result to $H$ -intersecting families for bounded-codegree $3$ -graphs. This is described as the first nontrivial improvement toward the conjectured maximum $2^{n-3}$ , but the claim is unverified here.

Current status (as of September 2026): A uniform gap below one half is claimed, while the sharp maximum $2^{n-3}$ remains open.

Sources

No solutions have been posted yet.