Problem 856

Erdős Problem #856: declared status 'open'. Formalized: no. Let $k\geq 3$ and $f_k(N)$ be the maximum value of $\sum_{n\in A}\frac{1}{n}$, where $A$ ranges over all subsets of $\{1,\ldots,N\}$ which contain no subset of size $k$ with the same pairwise least common multiple. Estimate $f_k(N)$. Current best: Improved bounds have been given by Tang and Zhang [TaZh25], who proved bounds of the shape$$(\log N)^{b_k-o(1)}\leq f_k(N)\leq (\log N)^{c_k+o(1)}$$for some constants $0<b_k\leq c_k\leq 1$, and in particular$$(\log N)^{0.438}\leq f_3(N)\leq (\log N)^{0.889},$$say, for all large $N$. For example, the exponents $c_k$ are $<1$ (and so the upper bound above is non-trivial) if and only if [857] holds for $k$-sunflowers. Prize: no. Tags: number theory.