An alternative proof of the upper bound for the generalised Erd\H{o}s box problem

In this article, we present an alternative proof of the classical theorem of Erd\H{o}s on the Tur\'{a}n numbers of complete $r$-partite $r$-uniform hypergraphs. More precisely, we establish that for finite sets $A_{1},\ldots,A_{r}$ with $|A_{1}|\leq\cdots\leq|A_{r}|$ and sufficiently large positive integer $n$, $$\mathrm{ex}(n,\mathbb{K}^{(r)}[A_{1},\ldots,A_{r}]) =O\left(n^{r-\frac{1}{|A_{1}|\cdots|A_{r-1}|}}\right).$$ Our approach develops a framework based on repeated applications of H\"older'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 Erd\H{o}s's classical upper bound and offers a unified perspective on the generalized Erd\H{o}s box problem.