Related papers: On maximal isolation sets in the uniform intersect…

A family $\mathcal{F} \subset \mathcal{P}(n)$ is $r$-wise $k$-intersecting if $|A_1 \cap \dots \cap A_r| \geq k$ for any $A_1, \dots, A_r \in \mathcal{F}$. It is easily seen that if $\mathcal{F}$ is $r$-wise $k$-intersecting for $r \geq 2$,…

A family $\mathcal{F}$ of subsets of $\{1,\dots,n\}$ is called $k$-wise intersecting if any $k$ members of $\mathcal{F}$ have non-empty intersection, and it is called maximal $k$-wise intersecting if no family strictly containing…

We use an entropy based method to study two graph maximization problems. We upper bound the number of matchings of fixed size $\ell$ in a $d$-regular graph on $N$ vertices. For $\frac{2\ell}{N}$ bounded away from 0 and 1, the logarithm of…

Clustering bipartite graphs is a fundamental task in network analysis. In the high-dimensional regime where the number of rows $n_1$ and the number of columns $n_2$ of the associated adjacency matrix are of different order, existing methods…

A joint of a set of lines $\mathcal{L}$ in $\mathbb{F}^d$ is a point that is contained in $d$ lines with linearly independent directions. The joints problem asks for the maximum number of joints that are formed by $L$ lines. Guth and Katz…

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…

Extremal problems related to the enumeration of graph substructures, such as independent sets, matchings, and induced matchings, have become a prominent area of research with the advancement of graph theory. A subset of vertices is called a…

For positive integers $n>k>t$ let $\binom{[n]}{k}$ denote the collection of all $k$-subsets of the standard $n$-element set $[n]=\{1,\ldots,n\}$. Subsets of $\binom{[n]}{k}$ are called $k$-graphs. A $k$-graph $\mathcal{F}$ is called…

Let $\mathcal{F}$ and $\mathcal{G}$ be two $t$-uniform families of subsets over $[k] = \{1,2,...,k\}$, where $|\mathcal{F}| = |\mathcal{G}|$, and let $C$ be the adjacency matrix of the bipartite graph whose vertices are the subsets in…

A subset $A$ of $[n] = \{1, \dots, n\}$ is $k$-separated if, when the elements of $[n]$ are considered on a circle, between any two elements of $A$ there are at least $k$ elements of $[n]$ that are not in $A$. A family $\mathcal{A}$ of sets…

Let $k, r, n \geq 1$ be integers, and let $\S_{n, k, r}$ be the family of $r$-signed $k$-sets on $[n] = \{1, \dots, n\}$ given by $$ \mathcal{S}_{n, k, r} = \Big\{\{(x_1, a_1), \dots, (x_k, a_k)\}: \{x_1, \dots, x_k\} \in \binom{[n]}{k},…

For two positive integers $k$ and $\ell$, a $(k \times \ell)$-spindle is the union of $k$ pairwise internally vertex-disjoint directed paths with $\ell$ arcs between two vertices $u$ and $v$. We are interested in the (parameterized)…

In this paper, we show how one may (efficiently) construct two types of extremal combinatorial objects whose existence was previously conjectural. (*) Panchromatic Graphs: For fixed integer k, a k-panchromatic graph is, roughly speaking, a…

In this paper, we obtain two spectral upper bounds for the $k$-independence number of a graph which is is the maximum size of a set of vertices at pairwise distance greater than $k$. We construct graphs that attain equality for our first…

Let $n(k_1, k_2)$ be the least integer $n$ such that there exists a graph on $n$ vertices in which every vertex is contained in both a clique of size $k_1$ and an independent set of size $k_2$. Recently, Feige and Pauzner showed that ${n(k,…

Consider families of $k$-subsets (or blocks) on a ground set of size $v$. Recall that if all $t$-subsets occur with the same frequency $\lambda$, one obtains a $t$-design with index $\lambda$. On the other hand, if all $t$-subsets occur…

Motivated by finding planar embeddings that lead to drawings with favorable aesthetics, we study the problems MINMAXFACE and UNIFORMFACES of embedding a given biconnected multi-graph such that the largest face is as small as possible and…

Given a sparse undirected graph G with weights on the edges, a k-plex partition of G is a partition of its set of nodes such that each component is a k-plex. A subset of nodes S is a k-plex if the degree of every node in the associated…

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…

The forbidden number forb$(m,F)$, which denotes the maximum number of distinct columns in an $m$-rowed $(0,1)$-matrix with no submatrix that is a row and column permutation of $F$, has been widely studied in extremal set theory. Recently,…