相关论文: Cross $t$-intersecting families for symplectic pol…

A family $\mathcal{F}$ is $t$-$\it{intersecting}$ if any two members have at least $t$ common elements. Erd\H os, Ko, and Rado proved that the maximum size of a $t$-intersecting family of subsets of size $k$ is equal to $ {{n-t} \choose…

A family of $k$-element subsets of an $n$-element set is called 3-wise intersecting if any three members in the family have non-empty intersection. We determine the maximum size of such families exactly or asymptotically. One of our results…

The profile vector of a family $\mathcal{F}$ of subsets of an $n$-element set is $(f_0,f_1, \ldots, f_n)$ where $f_i$ denotes the number of the $i$-element members of $\mathcal{F}$. In this paper we determine the extreme points of the set…

Two families $\mathcal{A}$ and $\mathcal{B}$ of sets are called cross-intersecting if each pair of sets $A\in \mathcal{A}$ and $B\in \mathcal{B}$ has nonempty intersection. Let $\cal{A}$ and ${\cal B}$ be two cross-intersecting families of…

Let $\mathcal{A}$ and $\matchcal{B}$ denote two families of subsets of an $n$-element set. The pair $(\mathcal{A},\mathcal{B})$ is said to be $\ell$-cross-intersecting iff $|A\cap B| = \ell$ for all $A\in\mathcal{A}$ and $B\in\mathcal{B}$.…

A finite classical polar space of rank $n$ consists of the totally isotropic subspaces of a finite vector space equipped with a nondegenerate form such that $n$ is the maximal dimension of such a subspace. A $t$-Steiner system in a finite…

Two families $\mathcal{F}$ and $\mathcal{G}$ are cross-intersecting if every set in $\mathcal{F}$ intersects every set in $\mathcal{G}$. The covering number $\tau(\mathcal{F})$ of a family $\mathcal{F}$ is the minimum size of a set that…

Let $\mathcal{S}_n$ be the symmetric group on the set $[n]:=\{1,2,\ldots,n\}$. A family $\mathcal{F}\subset \mathcal{S}_n$ is called intersecting if for every $\sigma,\pi\in \mathcal{F}$ there exists some $i\in [n]$ such that…

Let f: V --> U be a smooth non-isotrivial family of canonically polarized n-dimensional complex manifolds, where U is the complement of a normal crossing divisor S in a projective manifold Y. We show that some symmetric product of the sheaf…

We show that all maximal almost disjoint families have pseudocompact Vietoris hyperspace if and only if $\mathsf{MA}_\mathfrak c (\mathcal P(\omega)/\mathrm{fin})$ holds. We further study the question whether there is a maximal almost…

A family of perfect matchings of $K_{2n}$ is $t$-$intersecting$ if any two members share $t$ or more edges. We prove for any $t \in \mathbb{N}$ that every $t$-intersecting family of perfect matchings has size no greater than $(2(n-t) -…

We study symplectic structures on four-dimensional small covers. Our main result shows that every symplectic four-dimensional small cover is aspherical. We then classify symplectic small covers over products of two polygons, proving that…

Let $[n]$ (resp. $V$) be an $n$-element set (resp. $n$-dimensional vector space over the finite field $\mathbb{F}_{q}$), and $\binom{[n]}{k}$ (resp. $\genfrac{[}{]}{0pt}{}{V}{k}$) denote the set of all $k$-subsets of $[n]$ (resp.…

For a family $\mathcal{F}$ of subsets of a finite set, define $\mathcal{D}(\mathcal{F})=\{F\setminus F': F, F'\in\mathcal{F}\}$. A family $\mathcal{F}$ is called intersecting if $F\cap F'\not=\emptyset$ for all $F, F'\in\mathcal{F}$. Frankl…

Two sets $\mathscr{A}$ and $\mathscr{B}$ are said to be cross-intersecting if $X\cap Y\neq\emptyset$ for all $X\in\mathscr{A}$ and $Y\in\mathscr{B}$. Given two cross-intersecting Sperner families (or antichains) $\mathscr{A}$ and…

Two families $\mathcal{A}$ and $\mathcal{B}$, of $k$-subsets of an $n$-set, are {\em cross $t$-intersecting} if for every choice of subsets $A \in \mathcal{A}$ and $B \in \mathcal{B}$ we have $|A \cap B| \geq t$. We address the following…

A family of graphs F is said to be triangle-intersecting if for any two graphs G,H in F, the intersection of G and H contains a triangle. A conjecture of Simonovits and Sos from 1976 states that the largest triangle-intersecting families of…

A family of subsets of $[n]$ is intersecting if every pair of its sets intersects. Determining the structure of large intersecting families is a central problem in extremal combinatorics. Frankl-Kupavskii and Balogh-Das-Liu-Sharifzadeh-Tran…

Let $E_\lambda$ be the Legendre elliptic curve of equation $Y^2=X(X-1)(X-\lambda)$. We recently proved that, given $n$ linearly independent points $P_1(\lambda), \dots,P_n(\lambda)$ on $E_\lambda$ with coordinates in…

Let $L = \{\frac{a_1}{b_1}, \ldots , \frac{a_s}{b_s}\}$, where for every $i \in [s]$, $\frac{a_i}{b_i} \in [0,1)$ is an irreducible fraction. Let $\mathcal{F} = \{A_1, \ldots , A_m\}$ be a family of subsets of $[n]$. We say $\mathcal{F}$ is…