Related papers: Uniform Set Systems with Uniform Witnesses

A subset of a normed space $X$ is called equilateral if the distance between any two points is the same. Let $m(X)$ be the smallest possible size of an equilateral subset of $X$ maximal with respect to inclusion. We first observe that…

We consider families of $k$-subsets of $\{1, \dots, n\}$, where $n$ is a multiple of $k$, which have no perfect matching. An equivalent condition for a family $\mathcal{F}$ to have no perfect matching is for there to be a blocking set,…

The typical extremal problem asks how large a structure can be without containing a forbidden substructure. The Erd\H{o}s-Rothschild problem, introduced in 1974 by Erd\H{o}s and Rothschild in the context of extremal graph theory, is a…

In 1984, Wilson proved the Erd\H{o}s-Ko-Rado theorem for $t$-intersecting families of $k$-subsets of an $n$-set: he showed that if $n\ge(t+1)(k-t+1)$ and $\mathcal{F}$ is a family of $k$-subsets of an $n$-set such that any two members of…

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…

We study intersecting families of words from the Erd\H{o}s-Ko-Rado perspective. When the alphabet size is $2$, a maximum intersecting family is not necessarily a star. However, we prove that every maximum $3$-wise intersecting family is a…

The maximum size of an $r$-uniform hypergraph without a Berge cycle of length at least $k$ has been determined for all $k \ge r+3$ by F\"uredi, Kostochka and Luo and for $k<r$ (and $k=r$, asymptotically) by Kostochka and Luo. In this paper,…

The problem of bounding the size of a set system under various intersection restrictions has a central place in extremal combinatorics. We investigate the maximum number of disjoint pairs a set system can have in this setting. In…

Let $2^{[n]}$ denote the power set of $[n]:=\{1,2,..., n\}$. A collection $\B\subset 2^{[n]}$ forms a $d$-dimensional {\em Boolean algebra} if there exist pairwise disjoint sets $X_0, X_1,..., X_d \subseteq [n]$, all non-empty with perhaps…

The union-closed sets conjecture states that if a family of sets $\mathcal{A} \neq \{\emptyset\}$ is union-closed, then there is an element which belongs to at least half the sets in $\mathcal{A}$. In 2001, D. Reimer showed that the average…

We study dimensions of sumsets and iterated sumsets and provide natural conditions which guarantee that a set $F \subseteq \mathbb{R}$ satisfies $\overline{\dim}_\text{B} F+F > \overline{\dim}_\text{B} F$ or even $\dim_\text{H} n F \to 1$.…

Let $k$, $t$ and $m$ be positive integers. A $k$-multiset of $[m]$ is a collection of $k$ elements of $[m]$ with repetition and without ordering. We use $\left(\binom {[m]}{k}\right)$ to denote all the $k$-multisets of $[m]$. Two multiset…

In this note we disprove a conjecture of Kuzmin and Warmuth claiming that every family whose VC-dimension is at most d admits an unlabeled compression scheme to a sample of size at most d. We also study the unlabeled compression schemes of…

We prove that the `Upper Matching Conjecture' of Friedland, Krop, and Markstr\"om and the analogous conjecture of Kahn for independent sets in regular graphs hold for all large enough graphs as a function of the degree. That is, for every…

A $k$-uniform family $\mathcal{F}$ is called intersecting if $F\cap F'\neq \emptyset$ for all $F,F'\in \mathcal{F}$. The shadow family $\partial \mathcal{F}$ is the family of $(k-1)$-element sets that are contained in some members of…

Let $\{(A_i,B_i)\}_{i=1}^m$ be a set pair system. F\"{u}redi, Gy\'{a}rf\'{a}s and Kir\'{a}ly called it {\em $1$-cross intersecting} if $|A_i\cap B_j|$ is $1$ when $i\neq j$ and $0$ if $i=j$. They studied such systems and their…

For a field $\mathbb{F}$ and integers $d, k$ and $\ell$, a set $A \subseteq \mathbb{F}^d$ is called $(k,\ell)$-nearly orthogonal if all vectors in $A$ are non-self-orthogonal and every $k+1$ vectors in $A$ contain $\ell + 1$ pairwise…

Let $V$ be an $n$-dimensional vector space over the finite field $\mathbb{F}_{q}$ and let $\left[V\atop k\right]_q$ denote the family of all $k$-dimensional subspaces of $V$. A family $\mathcal{F}\subseteq \left[V\atop k\right]_q$ is called…

In 1960, Gr\"{u}nbaum proved that for any convex body $C\subset\mathbb{R}^d$ and every halfspace $H$ containing the centroid of $C$, one has that the volume of $H\cap C$ is at least a $\frac{1}{e}$-fraction of the volume of $C$. Recently,…

For an integer $d \geq 2$, a family $\mathcal{F}$ of sets is $\textit{$d$-wise intersecting}$ if for any distinct sets $A_1,A_2,\dots,A_d \in \mathcal{F}$, $A_1 \cap A_2 \cap \dots \cap A_d \neq \emptyset$, and $\textit{non-trivial}$ if…