Harmonic LCM patterns and sunflower-free capacity

TL;DR

This paper establishes new lower bounds on the sum of reciprocals of LCM-$k$-free sets and explores their connection to sunflower-free capacities, providing insights into Erdős's sunflower conjecture and related combinatorial structures.

Contribution

It introduces explicit lower bounds for LCM-$k$-free sets and links these bounds to sunflower-free capacities, advancing understanding of Erdős's sunflower problem.

Findings

Proves that $f_k(N)$ grows at least as fast as $(rac{ ext{log} N}{ ext{polylog} N})^{c_k}$.

Shows the connection between sunflower-free capacity and the growth rate of $f_k(N)$.

Demonstrates the sunflower conjecture's failure for fixed $k$ if and only if $f_k(N)$ is near $( ext{log} N)^{1-o(1)}$.

Abstract

Fix an integer $k\ge 3$. Call a set $A\subseteq [N]$ LCM-$k$-free if it does not contain distinct $a_1,\dots,a_k$ such that $\mathrm{lcm}(a_i,a_j)$ is the same for all $1\le i<j\le k$. Define $$ f_k(N):=\max\left\{\sum_{a\in A}\frac1a: A\subseteq [N] \text{ is LCM-$k$-free}\right\}. $$ Addressing a problem of Erd\H{o}s, we prove an explicit unconditional lower bound $$ f_k(N)\ge (\log N)^{c_k-o(1)}, \qquad c_k:=\frac{k-2}{e((k-2)!)^{1/(k-2)}}. $$ Let $F_k(n)$ denote the maximum size of a $k$-sunflower-free family of subsets of $[n]$, and define the Erd\H{o}s--Szemer\'edi $k$-sunflower-free capacity by $\mu_k^{\mathrm S}:=\limsup_{n\to\infty}F_k(n)^{1/n}$. Motivated by a remark of Erd\H{o}s relating this problem to the sunflower conjecture, we show that $$ (\log N)^{\log\mu_k^{\mathrm S}-o(1)} \le f_k(N) \ll (\log N)^{\mu_k^{\mathrm S}-1+o(1)}. $$ Furthermore, we show that the Erd\H{o}s--Szemer\'edi sunflower conjecture fails for this fixed $k$ (i.e. $\mu_k^{\mathrm S}=2$) if and only if $f_k(N)=(\log N)^{1-o(1)}$.