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Abstract:For integers $n\ge d+1$, let $\mathsf{M}_d(n)$ denote the maximum size of a $(d+1)$-uniform family on an $n$-element ground set with VC-dimension at most $d$. For $n\ge2d+2$, the classical construction of Ahlswede and Khachatrian, later generalized by Mubayi and Zhao, gives \[
\mathsf{M}_d(n)\ge \binom{n-1}{d}+\binom{n-4}{d-2}. \] We introduce a two-cover lifting construction and prove the recursive lower bound \[
\mathsf{M}_d(n)\ge
\binom{n-1}{d}+\binom{n-4}{d-2}+\mathsf{M}_{d-3}(n-5) \] for every $d\ge 3$ and $n\ge d+3$. Consequently, \[
\mathsf{M}_d(n)\ge
\binom{n-1}{d}+\binom{n-4}{d-2}+\binom{n-6}{d-3}. \] Thus the Mubayi--Zhao conjecture on the exact value of $\mathsf{M}_d(n)$ for $n\ge2(d+2)$ is false for any $d\ge 3$. The proof is elementary and proceeds entirely through an explicit analysis of traces.

Submission history

From: Xiaochen Zhao [view email]
[v1] Sat, 20 Jun 2026 14:26:23 UTC (6 KB)