Let P(n,k) be the set of integer partitions of n into exactly k positive parts. For 1≤ t<k, call a family A⊆P(n,k) t-intersecting if every lambda,mu∈A have at least t common parts, counted with multiplicity. Let S_t=lambda∈P(n,k):lambda contains at least t parts equal to 1 be the canonical t-star. Borg's conjecture asserts that every t-intersecting family satisfies |A|≤ |S_t|, equivalently that the canonical t-star is extremal.

Borg’s conjecture on intersecting integer partitions

Let $\mathcal{P}(n,k)$ be the set of integer partitions of $n$ into exactly $k$ positive parts. For $1\le t<k$ , call a family $\mathcal{A}\subseteq\mathcal{P}(n,k)$ $t$ -intersecting if every $\lambda,\mu\in\mathcal{A}$ have at least $t$ common parts, counted with multiplicity. Let $\mathcal{S}_t=\{\lambda\in\mathcal{P}(n,k):\lambda\text{ contains at least }t\text{ parts equal to }1\}$ be the canonical $t$ -star. Borg's conjecture asserts that every $t$ -intersecting family satisfies $|\mathcal{A}|\le |\mathcal{S}_t|$ , equivalently that the canonical $t$ -star is extremal.

References

Primary source

arXiv

Additional references

Intersecting integer partitions: star bounds and counterexamples at every scale — arXiv — Yury Person, Thomas Schweser

extremal-combinatorics

Posted 5 days ago · edited 5 days ago

Progress summary

Refreshed 5 days ago

Claimed progress

A new paper proves the conjecture in a broad large-size range and gives counterexamples elsewhere, so the full question remains open.

Borg (2013) conjectured that the family of partitions containing a prescribed number of common parts is extremal among intersecting partitions. The general extremal problem has both positive results and counterexamples, with intermediate cases unresolved.

Known results

For $t$ -intersection, the star is extremal when $n \le 2k-t+1$ or $n \ge 3tk^5$ (Borg, 2013).
In the large- $n$ range, the star is unique when $k \ge t+3$ (Borg, 2013).
Counterexamples to a universal fixed-length assertion include $(n,k)=(8,3)$ for $t=1$ , and more generally $(n,k)=(t+7,t+2)$ (Borg, 2013).

October 2026 star bounds and counterexamples

Yury Person and Thomas Schweser report a large- $n$ regime in which the star is extremal, together with counterexamples for every sufficiently large $k$ and each scale parameter $d$ . This narrows the unresolved region but does not settle the full conjecture; the paper’s claims are not independently verified in the retrieved material.

Current status (as of October 2026): Borg’s conjecture is proved in substantial parameter ranges and disproved as a universal threshold statement, while the remaining intermediate extremal cases are open.

Sources

No solutions have been posted yet.