Harmonic LCM patterns and sunflower-free capacity
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918260725776384 |
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| author | Tang, Quanyu Zhang, Shengtong |
| author_facet | Tang, Quanyu Zhang, Shengtong |
| contents | 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 Erdős, 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 Erdős--Szemerédi $k$-sunflower-free capacity by $μ_k^{\mathrm S}:=\limsup_{n\to\infty}F_k(n)^{1/n}$. Motivated by a remark of Erdős relating this problem to the sunflower conjecture, we show that $$ (\log N)^{\logμ_k^{\mathrm S}-o(1)} \le f_k(N) \ll (\log N)^{μ_k^{\mathrm S}-1+o(1)}. $$ Furthermore, we show that the Erdős--Szemerédi sunflower conjecture fails for this fixed $k$ (i.e. $μ_k^{\mathrm S}=2$) if and only if $f_k(N)=(\log N)^{1-o(1)}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_20055 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Harmonic LCM patterns and sunflower-free capacity Tang, Quanyu Zhang, Shengtong Number Theory Combinatorics Primary 11B75, Secondary 05D05, 11N25 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 Erdős, 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 Erdős--Szemerédi $k$-sunflower-free capacity by $μ_k^{\mathrm S}:=\limsup_{n\to\infty}F_k(n)^{1/n}$. Motivated by a remark of Erdős relating this problem to the sunflower conjecture, we show that $$ (\log N)^{\logμ_k^{\mathrm S}-o(1)} \le f_k(N) \ll (\log N)^{μ_k^{\mathrm S}-1+o(1)}. $$ Furthermore, we show that the Erdős--Szemerédi sunflower conjecture fails for this fixed $k$ (i.e. $μ_k^{\mathrm S}=2$) if and only if $f_k(N)=(\log N)^{1-o(1)}$. |
| title | Harmonic LCM patterns and sunflower-free capacity |
| topic | Number Theory Combinatorics Primary 11B75, Secondary 05D05, 11N25 |
| url | https://arxiv.org/abs/2512.20055 |