Let $\mathrm{Sym(n,k)}$ denote the set of permutations on $\{1,2,\ldots,n\}$ with exactly $k$ cycles. A family $\mathcal{F}\subset\mathrm{Sym}(n,k)$ is said to be intersecting if $\sigma^{-1}\tau$ has a fixed point for all $\sigma,\tau\in\mathcal{F}$. In this paper, we investigate the size and structure of maximum-sized intersecting families of permutations in $\mathrm{Sym}(n,k)$. In the regime $k\leq n^{0.25}$, we show that every maximum-sized intersecting family is a star, meaning it consists of all permutations in $\mathrm{Sym}(n,k)$ that agree at a given point in $[n]$. We establish this result by proving a stronger stability result that bounds the maximum possible size of a non-centred intersecting family. Specifically, in the regime $k\leq n^{0.25}$, the size of any non-centred intersecting family is at most $\left(2/3+o(1)\right)$ times the maximum possible size of a star. In the tighter polylogarithmic regime $k\leq (\ln n)^{d}$, we improve this bound to $\left(1-1/e+o(1)\right)$ times the maximum possible size of a star; we show that this bound is asymptotically sharp. Thus, we establish both an Erd\H{o}s--Ko--Rado theorem and its corresponding stability version for $\mathrm{Sym}(n,k)$.
A family $\mathcal{F}$ of subsets of $[n] := \{1,2,\ldots, n\}$ is called maximal $k$-wise intersecting if every collection of at most $k$ members of $\mathcal{F}$ has a non-empty intersection, and adding any other set to $\mathcal{F}$ breaks this property. An old question by Erd\H{o}s and Kleitman from 1974 asks for t...
Let $L$ be a fixed set of positive integers. A family $\mathcal{F}\subseteq 2^{[n]}$ is called $L$-differencing if $\lvert A\setminus B\rvert\in L$ for every ordered pair of distinct members $A,B\in\mathcal{F}$. A longstanding conjecture of Frankl, proposed in 1985, asserts that every $L$-differencing family has size a...
Let $f(N)$ denote the largest size of a set $A\subseteq [N]=\{1,\ldots,N\}$ containing no distinct $a,b,c$ such that \[ \frac2a=\frac1b+\frac1c . \] We prove \[ f(N)\gg N\exp\!\left(-(2\sqrt{\log(24/7)}+o(1))\sqrt{\log\log N}\right). \] The construction filters the odd integers up to $N$ by a random affine image of a d...
Let $H$ be a numerical monoid, that is, a cofinite submonoid of $\mathbb N$ (the non-negative integers under addition). Denote by $\mathcal P_{\text{fin},0}(H)$ the monoid obtained by endowing the family of all finite subsets of $H$ containing $0$ with the operation of setwise addition induced by $H$ on its power set....
Let $\widetilde{S}\to S$ be an unbranched regular $p$-fold cyclic cover of a closed orientable surface $S$ of genus $g$. Two natural groups are associated with this cover. The first is the centralizer in $\mathrm{Mod}(\widetilde{S})$ of a chosen generator $\sigma$ of the deck transformation group, denoted by $\mathrm{M...
For a graph $H$ and a family of graphs $\mathcal F$, let $\text{ex}(n,H,\mathcal F)$ denote the maximum number of copies of $H$ in an $\mathcal F$-free graph on $n$ vertices. For every integer $i\ge 3$, let $C_i$ denote the cycle of length $i$. For $r\ge 3$, set $\mathscr {C}_r=\{C_3,C_4,\ldots,C_r\},$ and set $\mathsc...
Zhen Liu, Chuan-Shu Wu· 0 citations
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