Star bound at the natural tk² threshold

Establish that there exists an absolute constant C>0 such that, for all sufficiently large k, every integer 1≤t<k, and every n≥Ctk², each t-intersecting family of integer partitions of n into exactly k positive parts has size at most p(n−t,k−t), the size of the canonical t-star.

Background

The paper studies t-intersecting families of integer partitions of n into exactly k positive parts, where common parts are counted with multiplicity. The canonical t-star consists of all partitions containing at least t copies of the part 1 and has size p(n−t,k−t).

Theorem 1 establishes the canonical-star upper bound when n≥Ak³ for every fixed A>24 and sufficiently large k, while the paper constructs counterexamples when n is of order tk²/3. These results show that a uniform star theorem requires a threshold of at least order tk², and motivate the conjecture that a sufficiently large constant multiple of tk² is sufficient.

References

We conjecture that a sufficiently large constant multiple of $tk2$ already suffices.

There is an absolute constant $C>0$ such that, for all sufficiently large $k$, all $1\le t<k$, and all $n\ge Ctk2$, every $t$-intersecting $\mathcal F\subseteq P_{n,k}$ satisfies

|\mathcal F|\le p(n-t,k-t).

— Intersecting integer partitions: star bounds and counterexamples at every scale  (2610.01747 - Person et al., 1 Oct 2026) in Concluding remarks, Conjecture 1 (labelled \ref{conj:all})