Harmonic LCM patterns and sunflower-free capacity

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 $$f_k(N):=\max \left\{\sum _{a\in A}\frac{1}{a}:A\subseteq [N]\text{ is LCM-k-free}\right\}.$$ Addressing a problem of Erdős, we prove an explicit unconditional lower bound $$f_k(N)\ge (\log N{)}^{c_k-o(1)},c_k:=\frac{k-2}{e((k-2)!{)}^{1/(k-2)}}.$$ Let $F_k(n)$ denote the maximum size of a $k$-sunflower-free family of subsets of $[n]$, and define the Erdős--Szemerédi $k$-sunflower-free capacity by ${\mu }_k^{\mathrm{S}}:={\mathrm{limsup}}_{n\to \infty }{F}_k(n{)}^{1/n}$. Motivated by a remark of Erdős 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 Erdős--Szemerédi sunflower conjecture fails for this fixed $k$ (i.e. ${\mu }_k^{\mathrm{S}}=2$) if and only if $f_k(N)=(\log N{)}^{1-o(1)}$.

19 pages. Comments and suggestions are welcome