Year: 2026
Type: preprint
Abstract: Fix $k\ge 3$, and let $f_k(N)$ be the largest harmonic sum of a subset of $[N]$ containing no $k$ distinct integers with a common pairwise least common multiple. We prove that $f_k(N)=(\log N)^{γ_k+o(1)}$ for a well-defined exponent $γ_k\in(0,1]$. Following the weighted-pressure idea of Chojecki, we give a self-contained proof of variational formulas for $γ_k$ in terms of weighted sunflower-free families. We then eliminate the continuous weight: if $M_k(n,r)$ is the largest size of an $r$-uniform $k$-cosunflower-free family on $[n]$, then \[ γ_k=\sup_{n\ge1,\ 1\le r\le n}\frac{r}{en}M_k(n,r)^{1/r}. \] This finite-block formula yields a direct transfer principle from uniform set-system constructions, recovers the Tang--Zhang bounds, and gives $0.438899\ldots<γ_3\le0.889881\ldots$.
Source: arXiv (Cornell University)
Authors: Luo, Yanping, Yang, Ruiyi, Zhu, Keheng
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7212059601