The study of $t$-intersecting families in symmetric groups received lots of attention in the last two decades. In this paper, we study two different kinds of stability results for $t$-intersecting families in symmetric groups. Let $\Sigma_n$ denote the symmetric group on $\{1,2,\ldots,n\}$. The $t$-diversity $\gamma_t(...
Yu-Hang Cao, Gen-Nian Ge, Jian Wang et al.· 0 citations
For fixed $d\geq 2$, let $\tau_d(n)$ be the minimum size of a set $S\subseteq\{0,\ldots,n\}^d$ such that the affine lines determined by pairs of distinct points of $S$ cover the grid. Let $\sigma_d(n)$ be the analogous minimum when every grid point must lie on the closed segment joining two distinct points of $S$. A ce...
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...
