Related papers: Transversals to colorful intersecting convex sets

The colorful Helly theorem and Tverberg's theorem are fundamental results in discrete geometry. We prove a theorem which interpolates between the two. In particular, we show the following for any integers $d \geq m \geq 1$ and $k$ a prime…

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…

Let $\mathcal{F}$ be a family of convex sets in ${\mathbb R}^d$, which are colored with $d+1$ colors. We say that $\mathcal{F}$ satisfies the Colorful Helly Property if every rainbow selection of $d+1$ sets, one set from each color class,…

Let $\F$ be a finite family of axis-parallel boxes in $\R^d$ such that $\F$ contains no $k+1$ pairwise disjoint boxes. We prove that if $\F$ contains a subfamily $\M$ of $k$ pairwise disjoint boxes with the property that for every $F\in \F$…

For a positive integer $d\geq 2$, a family $\mathcal F\subseteq \binom{[n]}{k}$ is said to be d-wise intersecting if $|F_1\cap F_2\cap \dots\cap F_d|\geq 1$ for all $F_1, F_2, \dots ,F_d\in \mathcal F$. A d-wise intersecting family…

In this paper, we address several intersection problems for $r$-cross $t$-intersecting families of partitions. A $k$-partition of an $n$-set $X$ is a set of $k$ pairwise disjoint non-empty subsets whose union is $X$. For $1\leq i\leq r$,…

Two families $\mathcal{A}$ and $\mathcal{B}$ of sets are said to be cross-intersecting if each member of $\mathcal{A}$ intersects each member of $\mathcal{B}$. For any two integers $n$ and $k$ with $0 \leq k \leq n$, let ${[n] \choose \leq…

We propose a notion of depth with respect to a finite family $\mathcal{F}$ of convex sets in $\mathbb{R}^d$ which we call $\text{dep}_\mathcal{F}$. We begin showing that $\text{dep}_\mathcal{F}$ satisfies some expected properties for a…

We prove that for any set $F$ of $n\ge 2$ pairwise disjoint open convex sets in $\mathbb{R}^3$, the connected components of the set of lines intersecting every member of $F$ are contractible. The same result holds for directed lines.

Given integers $r\geq 2$ and $n,t\geq 1$ we call families $\mathcal{F}_1,\dots,\mathcal{F}_r\subseteq\mathscr{P}([n])$ $r$-cross $t$-intersecting if for all $F_i\in\mathcal{F}_i$, $i\in[r]$, we have $\vert\bigcap_{i\in[r]}F_i\vert\geq t$.…

Ellis, Filmus, and Friedgut proved an old conjecture of Simonovits and S\'os showing that the maximum size of a triangle-intersecting family of graphs on $n$ vertices has size at most $2^{\binom{n}{2} - 3}$, with equality for the family of…

The classical Ham Sandwich theorem states that any $d$ point sets in $\mathbb{R}^d$ can be simultaneously bisected by a single affine hyperplane. A generalization of Dolnikov asserts that any $d$ families of pairwise intersecting compact,…

A family of sets $\mathcal{A}$ is union-closed if it is finite and nonempty with member sets that are all finite and distinct (at least one of which is nonempty) and it satisfies the property $X, Y \in \mathcal{A} \implies X \cup Y \in…

A family of sets has the $(p, q)$ property if among any $p$ members of it some $q$ intersect. It is shown that if a finite family of compact convex sets in $\R^2$ has the $(p+1,2)$ property then it is pierced by $\lfloor \frac{p}{2} \rfloor…

We prove that if two families $\mathcal{F} \subseteq \binom{[n]}{k}$ and $\mathcal{F}' \subseteq \binom{[n]}{k'}$ satisfy $\sum_{1 \leq i, j \leq \ell} \lvert F_i \cap F_j' \rvert \geq \ell^2t - \ell +1$ for every choice of distinct $F_1,…

We prove the following the generalized Tur\'an type result. A collection $\mathcal{T}$ of $r$ sets is an $r$-triangle if for every $T_1,T_2,\dots,T_{r-1}\in \mathcal{T}$ we have $\cap_{i=1}^{r-1}T_i\neq\emptyset$, but $\cap_{T\in…

Let $m\geq 2$, $n$ be positive integers, and $R_i=\{k_{i,1} >k_{i,2} >\cdots> k_{i,t_i}\}$ be subsets of $[n]$ for $i=1,2,\ldots,m$. The families $\mathcal{F}_1\subseteq \binom{[n]}{R_1},\mathcal{F}_2\subseteq…

Let $V$ be an $n$-dimensional vector space over the finite field $\mathbb{F}_q$, and ${V\brack k}$ denote the family of all $k$-dimensional subspaces of $V$. The families $\mathcal{F}_1\subseteq{V\brack k_1},\mathcal{F}_2\subseteq{V\brack…

A family of subsets $\mathcal{F}$ is intersecting if $A \cap B \neq \emptyset$ for any $A, B \in \mathcal{F}$. In this paper, we show that for given integers $k > d \ge 2$ and $n \ge 2k+2d-3$, and any intersecting family $\mathcal{F}$ of…

Let $G$ and $H$ be finite-dimensional vector spaces over $\mathbb{F}_p$. A subset $A \subseteq G \times H$ is said to be transverse if all of its rows $\{x \in G \colon (x,y) \in A\}$, $y \in H$, are subspaces of $G$ and all of its columns…