Maximum Intersection Order for Asymptotic $K(d)$-Families

Definitions and mathematical statement

Fix the intended positive integer $k$. The source asks for the largest parameter $d=d(k)$ for which every admissible $K(d)$-family $\mathcal G$ satisfies \[ |\mathcal G|\leq(1+o(1))\binom{n-1}{k-1} \quad\text{as }n\to\infty, \] and then asks for an exact extremal extension. The defining property $K(d)$, the allowed range of $d$ and the domain of the set family are not supplied. Although the binomial normalization suggests $k$-uniform subsets of an $n$-element set, that interpretation is not used to guess a $d$-wise intersection or forbidden-configuration condition.

Denote the missing admissible class by $\mathfrak K_{n,k,d}$. For fixed $k,d$, the uniform extremal meaning of the asymptotic bound is \[ \forall\varepsilon>0\ \exists n_0\ \forall n\geq n_0 \ \forall\mathcal G\in\mathfrak K_{n,k,d},\qquad |\mathcal G|\leq(1+\varepsilon)\binom{n-1}{k-1}. \] Thus the error may depend on fixed $k,d$ but must be uniform over families of that size. An error term chosen independently for each changing family would not express the intended maximum-size bound. The question seeks the largest admissible $d$ with this property; neither finiteness of the eligible range nor monotonicity in $d$ is stated, so existence of a finite maximum is not assumed in advance. The parameter $k$, and then the tested $d$, are fixed before taking $n\to\infty$.

A natural exact strengthening would replace $1+o(1)$ by $1$, for the intended finite or sufficiently-large range of $n$, and determine equality cases if required by "exact extremal result." The source does not specify that range, a sharp threshold $n_0(k,d)$, or extremal constructions. For example a full $k$-uniform star has size $\binom{n-1}{k-1}$, but without $K(d)$'s definition it is not asserted to be admissible or extremal. No exact characterization is inferred from an asymptotic numerical inequality alone.

The literature triage checked 2026-08-21 reports no authoritative complete resolution of the extracted claim located in its search. The open label and reported official-source verification remain supplied metadata; they do not recover the missing family condition or provide an independently checked current theorem.

Short English statement

Find the largest fixed $d$ allowing the stated uniform asymptotic size bound, then establish the intended exact bound and extremal cases. The decisive $K(d)$-family condition and finite-size regime are absent, so neither an intersection definition nor a star extremizer is guessed.

Sources

• Local UnsolvedMath registry, permanent ID 30000605, source code catalogue entry, "Maximum Intersection Order for Asymptotic $K(d)$-Families," including original and cleaned asymptotic/exact requests and verification metadata.
• Combinatorics, Probability and Computing (2007), Oberwolfach Reports, pp. 2875--2952, DOI 10.4171/owr/2006/48: https://doi.org/10.4171/owr/2006/48. This is also the sole supplied literature reference. The citation uses 2007, while the DOI and proposed-year metadata use 2006.
• Literature triage checked 2026-08-21; local record updated 2026-09-16. No external source has been visited or independently checked here.