For every nonnegative integer t, every integer m≥ 2, and every family of pairs of finite sets (A_i,B_i)_i=1^m satisfying |A_i cap B_i|≤ t for all i and |A_i cap B_j|>t for all i≠ j, one has displaystyle∑_i=1^m binom|A_i|+|B_i|-2t|A_i|-t^-1≤ 1.

Füredi’s conjecture

For every nonnegative integer $t$ , every integer $m\ge 2$ , and every family of pairs of finite sets $(A_i,B_i)_{i=1}^m$ satisfying $|A_i\cap B_i|\le t$ for all $i$ and $|A_i\cap B_j|>t$ for all $i\ne j$ , one has $\displaystyle\sum_{i=1}^m\binom{|A_i|+|B_i|-2t}{|A_i|-t}^{-1}\le 1$ .

References

Primary source

arXiv

Additional references

A Proof of Füredi's Conjecture — arXiv — Zihao Huang, Suijie Wang

algebraic-combinatorics extremal-set-theory

Posted 3 weeks ago · edited 3 weeks ago

Progress summary

Refreshed 3 weeks ago

Claimed solved

An unrefereed September 2026 preprint claims to prove Füredi’s conjecture, but independent verification has not yet been reported.

Füredi’s conjecture is a central extension of the Bollobás set-pairs inequality for strong Bollobás systems.

Known results

A 2024 preprint proved the conjecture when all pairs satisfy $|A_i|+|B_i|=N$ , and established further monotone-dimension and monotone-size special cases.

September 2026 claimed proof

Zihao Huang and Suijie Wang report a proof using a weighted subspace inequality and an exterior-ideal argument. If correct, this settles the full conjecture; the preprint is unrefereed.

Current status (as of September 2026): A full proof has been claimed by Huang and Wang, while the conjecture remains unverified pending refereeing or independent confirmation.

Sources

No solutions have been posted yet.