Related papers: Maximum shattering

We obtain new bounds for (a variant of) the Furstenberg set problem for high dimensional flats over $\mathbb{R}^n$. In particular, let $F\subset \mathbb{R}^n$, $1\leq k \leq n-1$, $s\in (0,k]$, and $t\in (0,k(n-k)]$. We say that $F$ is a…

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…

The Galvin problem asks for the minimum size of a family $\mathcal{F} \subseteq \binom{[n]}{n/2}$ with the property that, for any set $A$ of size $\frac n 2$, there is a set $S \in \mathcal{F}$ which is balanced on $A$, meaning that $|S…

A central theorem in combinatorics is Sperner's Theorem, which determines the maximum size of a family $\mathcal{F}\subseteq \mathcal{P}(n)$ that does not contain a $2$-chain $F_1\subsetneq F_2$. Erd\H{o}s later extended this result and…

We study the problem of determining the minimum number $f(n,k,d)$ of affine subspaces of codimension $d$ that are required to cover all points of $\mathbb{F}_2^n\setminus \{\vec{0}\}$ at least $k$ times while covering the origin at most…

Finding the maximum number of maximal independent sets in an $n$-vertex graph $G$, $i(G)$, from a restricted class is an extensively studied problem. Let $kK_2$ denote the matching of size $k$, that is a graph with $2k$ vertices and $k$…

Separating hash families are useful combinatorial structures which generalize several well-studied objects in cryptography and coding theory. Let $p_t(N, q)$ denote the maximum size of universe for a $t$-perfect hash family of length $N$…

We explore from several perspectives the following question: given $X\subseteq \mathbb{Z}$ and $N\in \mathbb{N}$, what is the maximum size $D(X,N)$ of $A\subseteq \{1,2,\dots,N\}$ before $A$ is forced to contain two distinct elements that…

Motivated by odd-sunflowers, introduced recently by Frankl, Pach, and P{\'a}lv{\"o}lgyi, we initiate the study of temperate families: a family $\mathcal{F} \subseteq \mathcal{P}([n])$ is said to be \emph{temperate} if each $A \in…

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 $F\in \F$…

A family $\mathcal{F}\subseteq\mathcal{P}(n)$ is an $(a,b)$-town$\pmod k$ if all sets in it have cardinality $a\pmod k$ and all pairwise intersections in it have cardinality $b\pmod k$. For $k=2$ the maximal size of such a family is known…

Let $\mathcal F\subset 2^{[n]}$ be an $s$-uniform family such that every two distinct sets have a nonempty intersection but intersect in at most $k$ elements. By the well-known Ray-Chaudhuri--Wilson theorem, since the intersections can take…

Let $\F\subset 2^{[n]}$ be a family of subsets of $\{1,2,..., n\}$. For any poset $H$, we say $\F$ is $H$-free if $\F$ does not contain any subposet isomorphic to $H$. Katona and others have investigated the behavior of $\La(n,H)$, which…

The Frankl conjecture, also known as the union-closed sets conjecture, states that in any finite non-empty union-closed family, there exists an element in at least half of the sets. From an optimization point of view, one could instead…

Let $n > k > t \geq j \geq 1$ be integers. Let $X$ be an $n$-element set, ${X\choose k}$ the collection of its $k$-subsets. A family $\mathcal F \subset {X\choose k}$ is called $t$-intersecting if $|F \cap F'| \geq t$ for all $F, F' \in…

Let $\mathscr{H}$ be a family of digraphs. A digraph $D$ is \emph{$\mathscr{H}$-free} if it contains no isomorphic copy of any member of $\mathscr{H}$. For $k\geq2$, we set $C_{\leq k}=\{C_{2}, C_{3},\ldots,C_{k}\}$, where $C_{\ell}$ is a…

The trace of a family of sets $\mathcal{A}$ on a set $X$ is $\mathcal{A}|_X=\{A\cap X:A\in \mathcal{A}\}$. If $\mathcal{A}$ is a family of $k$-sets from an $n$-set such that for any $r$-subset $X$ the trace $\mathcal{A}|_X$ does not contain…

We let $\mathcal{F}$ be a finite family of sets closed under taking unions and $\emptyset \not \in \mathcal{F}$, and call an element abundant if it belongs to more than half of the sets of $\mathcal{F}$. In this notation, the classical…

For a family $\mathcal{F}\subset \binom{[n]}{k}$ and two elements $x,y\in [n]$ define $\mathcal{F}(\bar{x},\bar{y})=\{F\in \mathcal{F}\colon x\notin F,\ y\notin F\}$. The double-diversity $\gamma_2(\mathcal{F})$ is defined as the minimum of…

Let $\mathcal{F}\subset\binom{[n]}{k}$ be an intersecting family. For an element $i\in[n]$, the degree of $i$ is the number of sets in $\mathcal{F}$ that contain $i$. Assume that the degrees are ordered as $d_{1}\ge d_{2}\ge\cdots\ge…