2026/07/18 by Subhankar Dash, Kaushik Majumder
Mathematics · #math.CO
In this article, we present an alternative proof of the classical theorem of Erdős on the Turán numbers of complete r-partite r-uniform hypergraphs. More precisely, we establish that for finite sets A1,…,Ar with |A1|≤⋯≤|Ar| and sufficiently large positive integer n, ex(n,\mathbbK(r)[A1,…,Ar]) =O(n^r-\frac1|A1|⋯|Ar-1|). Our approach develops a framework based on repeated applications of Hölder's inequality and the enumeration of configurations through multiple sums. The method combines the principle of inclusion--exclusion with a discrete analogue of Fubini's theorem, yielding recursive estimates for extremal quantities. This provides an alternative proof of Erdős's classical upper bound and offers a unified perspective on the generalized Erdős box problem.