An alternative proof of the upper bound for the generalised Erdős box problem

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 $A_1,\dots ,A_r$ with $\mathrm{\mid }{A}_1\mathrm{\mid }\le \cdots \le \mathrm{\mid }{A}_r\mathrm{\mid }$ and sufficiently large positive integer $n$, $$\mathrm{e}\mathrm{x}(n,{\mathbb{K}}^{(r)}[A_1,\dots ,A_r])=O\left(n^{r-\frac{1}{\mathrm{\mid }{A}_1\mathrm{\mid }\cdots \mathrm{\mid }{A}_{r-1}\mathrm{\mid }}}\right)\mathrm{.}$$ 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.