Related papers: Forbidden sparse intersections

Let $n > k > 1$ be integers, $[n] = \{1, \ldots, n\}$. Let $\mathcal F$ be a family of $k$-subsets of~$[n]$. The family $\mathcal F$ is called intersecting if $F \cap F' \neq \emptyset$ for all $F, F' \in \mathcal F$. It is called almost…

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,…

Let $\mathcal{F},\mathcal{G}$ be two cross-intersecting families of $k$-subsets of $\{1,2,\ldots,n\}$. Let $\mathcal{F}\wedge \mathcal{G}$, $\mathcal{I}(\mathcal{F},\mathcal{G})$ denote the families of all intersections $F\cap G$ with $F\in…

A family $\mathcal{F}$ of subsets of $\{1,2,\ldots,n\}$ is called a $t$-intersecting family if $|F\cap G| \geq t$ for any two members $F, G \in \mathcal{F}$ and for some positive integer $t$. If $t=1$, then we call the family $\mathcal{F}$…

Two families $\mathcal A\subseteq\binom{[n]}{k}$ and $\mathcal B\subseteq\binom{[n]}{\ell}$ are called cross-$t$-intersecting if $|A\cap B|\geq t$ for all $A\in\mathcal A$, $B\in\mathcal B$. Let $n$, $k$ and $\ell$ be positive integers such…

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 $B\in\mathcal{B}$.…

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 investigate the product measures of intersection problems in extremal combinatorics. Invoking a recent result of He--Li--Wu--Zhang, we prove that for any $ n \geq t \geq 3$ and $ p_1, p_2 \in (0, \frac{1}{t+1})$, if $ \mathcal{F}_1,…

A family $F$ of graphs on a fixed set of $n$ vertices is called triangle-intersecting if for any $G_1,G_2 \in F$, the intersection $G_1 \cap G_2$ contains a triangle. More generally, for a fixed graph $H$, a family $F$ is $H$-intersecting…

We call a family $\mathcal{F}$ $(3,2,\ell)$-intersecting if $|A \cap B|+|B \cap C|+|C \cap A| \geq \ell$ for all $A$, $B$, $C \in \mathcal{F}$. We try to look for the maximum size of such a family $\mathcal{F}$ in case when $\mathcal{F}…

We prove the following, for a universal constant $c>0$. Let $n \in \mathbb{N}$ and $1 \leq t<c\frac{n}{\log n}$. Let $F,G \subset S_n$ be families of permutations such that no $\sigma \in F$ and $\tau \in G$ agree on exactly $t-1$ values.…

Given a sequence of positive integers $p = (p_1, . . ., p_n)$, let $S_p$ denote the family of all sequences of positive integers $x = (x_1,...,x_n)$ such that $x_i \le p_i$ for all $i$. Two families of sequences (or vectors), $A,B \subseteq…

Let $\mathcal{A}=\{A_{1},...,A_{p}\}$ and $\mathcal{B}=\{B_{1},...,B_{q}\}$ be two families of subsets of $[n]$ such that for every $i\in [p]$ and $j\in [q]$, $|A_{i}\cap B_{j}|= \frac{c}{d}|B_{j}|$, where $\frac{c}{d}\in [0,1]$ is an…

Let M be a subset of {0, .., n} and F be a family of subsets of an n element set such that the size of A intersection B is in M for every A, B in F. Suppose that l is the maximum number of consecutive integers contained in M and n is…

Let $n$, $k$ and $t$ be positive integers, and let $\mathcal{F}$ be a collection of $k$-subsets of $[n]=\{1,2,\dots,n\}$. The $t$-covering number $\tau_t(\mathcal{F})$ of $\mathcal{F}$ is defined as the minimum size of a set $T$ such that…

A graph $G$ has $p$-intersection number at most $d$ if it is possible to assign to every vertex $u$ of $G$, a subset $S(u)$ of some ground set $U$ with $|U|=d$ in such a way that distinct vertices $u$ and $v$ of $G$ are adjacent in $G$ if…

For $n > 2k \geq 4$ we consider intersecting families $\mathcal F$ consisting of $k$-subsets of $\{1, 2, \ldots, n\}$. Let $\mathcal I(\mathcal F)$ denote the family of all distinct intersections $F \cap F'$, $F \neq F'$ and $F, F'\in…

Let $n$, $r$, $k_1,\ldots,k_r$ and $t$ be positive integers with $r\geq 2$, and $\mathcal{F}_i\ (1\leq i\leq r)$ a family of $k_i$-subsets of an $n$-set $V$. The families $\mathcal{F}_1,\ \mathcal{F}_2,\ldots,\mathcal{F}_r$ are said to be…

Let $n$, $r$, and $k$ be positive integers such that $k, r \geq 2$, $L$ a non-empty subset of $[k]$, and $\mathcal{F}_i \subseteq \binom{[n]}{k}$ for $1 \leq i \leq r$. We say that non-empty families $\mathcal{F}_1, \mathcal{F}_2, \ldots,…

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…