Balogh–Garcia–Li–Wagner conjectures on typical intersecting families

Let $\mathcal{I}(n,k)$ denote the set of intersecting families $\mathcal{F}\subseteq\binom{[n]}{k}$ , and let a family be trivial if all its members contain a common element. Then, as $k\to\infty$ , almost every member of $\mathcal{I}(n,k)$ is trivial for every $n\ge 2k+2$ ; in particular, for $n=2k+2$ , $|\mathcal{I}(2k+2,k)|=(2k+2+o(1))2^{\binom{2k+1}{k-1}}$ . The same conjecture proposed that, when $n=2k+1$ , almost every intersecting family is close to a full star, with the members outside the star forming components of size at most two in the graph in which two $k$ -sets are adjacent when they intersect in $k-1$ elements.

References

Primary source

arXiv

Additional references

Typical intersecting families at n=2k+1 and n=2k+2 — arXiv — Lina Li

extremal-combinatorics graph-containers probabilistic-combinatorics

Posted 5 days ago · edited 5 days ago

Progress summary

Refreshed 5 days ago

Claimed progress

A recent preprint settles the conjecture when the ground set is sufficiently larger than twice the set size, but the two critical boundary cases remain unresolved in the available evidence.

The conjectures describe which intersecting families of $k$ -sets are typical near the threshold $n=2k+2$ . The original work also proposed a separate near-star classification and enumeration at $n=2k+1$ .

Known results

Balogh, Garcia, Li, and Wagner (2021; revised 2024) proved typical triviality for $n\geq 2k+100\ln k$ .
Their conjecture extends typical triviality to every $n\geq 2k+2$ .
At $n=2k+1$ , they conjectured a distinct near-star structure and asymptotic enumeration.

2026 range extension

A 2026 arXiv preprint proves that non-trivial families are negligible for every $n\geq 2k+3$ as $k\to\infty$ , confirming the Balogh–Garcia–Li–Wagner conjecture throughout that range. The preprint leaves $n=2k+2$ unresolved and does not establish the $n=2k+1$ near-star conjecture.

Current status (as of October 2026): The conjecture is established for $n\geq 2k+3$ , while the cases $n=2k+2$ and the proposed near-star classification at $n=2k+1$ remain open in the independently supported record.

Sources

No solutions have been posted yet.