Extremal Bounds for Intersecting $K(d)$-Families

Definitions and mathematical statement

Let $k,d,n$ be integers with $3\leq d\leq k$ and $(d-1)n\geq dk$, equivalently $n\geq\lceil dk/(d-1)\rceil$. Write $[n]=\{1,\ldots,n\}$ and $\binom{[n]}k$ for its $k$-element subsets. A family $\mathcal G\subseteq\binom{[n]}k$ is a $K(d)$-family if \[ \forall A_1,\ldots,A_d\in\mathcal G\text{ pairwise distinct}:\quad \left|\bigcup_{i=1}^d A_i\right|\leq2k \ \Longrightarrow\ \bigcap_{i=1}^d A_i\ne\varnothing. \] This condition concerns exactly $d$ distinct members with small union; it is not silently replaced by pairwise intersection or by $d$-wise intersection with no union restriction. Equality at union size $2k$ is included.

The proposed extremal statement, throughout the stated integer range, is \[ |\mathcal G|\leq\binom{n-1}{k-1}, \] with equality only when there is $x\in[n]$ such that \[ \mathcal G=\{A\in\tbinom{[n]}k:x\in A\}. \] This is the full star at a common point. Such a star satisfies the condition and has the proposed cardinality. If equality is phrased only as having a common point, its cardinality already forces the full star; no maximality hypothesis is added. No strict inequality in the threshold, large-$n$ qualification, or unspecified exceptional cases are inserted.

The supplied literature triage checked 21 August 2026 reports no authoritative complete resolution of the extracted claim in its targeted search. It supplies no theorem-author attribution, case list or endpoint refinement; the dated open label is retained without independent current-status certification.

Short English statement

A $k$-uniform family in which any $d$ distinct sets with union of size at most $2k$ intersect should have at most one full star's size for $n\geq dk/(d-1)$. At equality it should be exactly a full star. Both the small-union condition and the complete threshold range are preserved.

Sources

• Local UnsolvedMath registry, permanent ID 30000604, exact source code catalogue entry; Extremal Bounds for Intersecting $K(d)$-Families. Proposed year 2006, status open; corrected_verified statement retains $k\geq d\geq3$, the $2k$ union threshold, size bound and common-point equality case.
• Supplied report citation: Combinatorics, Probability and Computing (2007), Oberwolfach Reports, DOI 10.4171/owr/2006/48, pp. 2875--2952, https://doi.org/10.4171/owr/2006/48. Title year differs from DOI/proposed year.
• Supplied literature triage checked 2026-08-21 cites the same report, https://doi.org/10.4171/owr/2006/48, and reports no authoritative complete resolution found. The dated OWR-derived curation requests independent verification of time-sensitive status and cited claims; no separate theorem/case reference is supplied.