Recursive Lifting Beyond the Ahlswede--Khachatrian Construction

Abstract

For the Erdős--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.