An Analytic counting Framework for the generalised Erd\H{o}s box problem

Subhankar Dash; Kaushik Majumder

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arXiv · 2607.16694

An Analytic counting Framework for the generalised Erd\H{o}s box problem

Subhankar Dash·Kaushik Majumder

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Abstract

In this article, we develop an $r-$uniform analogue of the classical K\"{o}v\'{a}ri--S\'{o}s--Tur\'{a}n inequality. This yields an alternative proof of Erd\H{o}s's classical upper bound for complete $r$-partite $r$-uniform hypergraphs. More precisely, we establish that for finite non-empty sets $A_{1},\ldots,A_{r}$ with $|A_{1}|\leq\cdots\leq|A_{r}|$ and sufficiently large positive integer $n$, \[\mathrm{ex}(n,\mathbb{K}^{(r)}[A_{1},\ldots,A_{r}])=O\left(n^{r-\frac{1}{|A_{1}|\ldots|A_{r-1}|}}\right).\] Our proof develops an analytic counting framework based on repeated applications of H\"{o}lder's inequality and the enumeration of configurations through multiple sums. The argument combines the principle of inclusion--exclusion with a discrete analogue of Fubini's theorem to obtain recursive estimates for extremal quantities. This provides a unified analytic perspective on the generalized Erd\H{o}s box problem.

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Subhankar Dash, Kaushik Majumder. 2026-07-18. An Analytic counting Framework for the generalised Erd\H{o}s box problem. https://arxiv.org/abs/2607.16694

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