Recursive Lifting Beyond the Ahlswede--Khachatrian Construction

Summary · qwen2.5:32b

The article presents a recursive lifting method that surpasses the Ahlswede--Khachatrian/Mubayi--Zhao construction for uniform set systems of bounded VC-dimension, specifically proving improved lower bounds for \(M_d(n)\) in every dimension \(d \ge 3\). This advancement is significant as it provides a new benchmark for the Erdős--Frankl--Pach problem. A concrete detail is the derived inequality: \[ M_d(n)\ge \binom{n-1}{d}+\binom{n-4}{d-2}+M_{d-3}(n-5) \] for \(n\ge d+3\).

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Excerpt

arXiv:2607.04858v1 Announce Type: new Abstract: 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.