Recursive Lifting Beyond the Ahlswede--Khachatrian Construction

For the Erd\H{o}s--Frankl--Pach problem on uniform set systems of bounded VC-dimension, the Ahlswede--Khachatrian/Mubayi--Zhao construction has long served as the standard lower-bound benchmark. We develop a recursive lifting method that goes beyond this benchmark in every dimension $d\ge3$, proving that for every $d\ge3$ and $n\ge d+3$, $$M_d(n)\ge \binom{n-1}{d}+\binom{n-4}{d-2}+M_{d-3}(n-5).$$ The proof is elementary and proceeds through explicit trace obstructions. We also record a further recursive improvement in the concluding remarks.