Recursive lower bounds for uniform set systems of bounded VC-dimension

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 $n\ge 2d+2$, the classical construction of Ahlswede and Khachatrian, later generalized by Mubayi and Zhao, gives $${\mathsf{M}}_d(n)\ge \left(\genfrac{}{}{0px}{}{n-1}{d}\right)+\left(\genfrac{}{}{0px}{}{n-4}{d-2}\right)\mathrm{.}$$ We introduce a two-cover lifting construction and prove the recursive lower bound $${\mathsf{M}}_d(n)\ge \left(\genfrac{}{}{0px}{}{n-1}{d}\right)+\left(\genfrac{}{}{0px}{}{n-4}{d-2}\right)+{\mathsf{M}}_{d-3}(n-5)$$ for every $d\ge 3$ and $n\ge d+3$. Consequently, $${\mathsf{M}}_d(n)\ge \left(\genfrac{}{}{0px}{}{n-1}{d}\right)+\left(\genfrac{}{}{0px}{}{n-4}{d-2}\right)+\left(\genfrac{}{}{0px}{}{n-6}{d-3}\right)\mathrm{.}$$ Thus the Mubayi–Zhao conjecture on the exact value of ${\mathsf{M}}_d(n)$ for $n\ge 2(d+2)$ is false for any $d\ge 3$. The proof is elementary and proceeds entirely through an explicit analysis of traces.