Related papers: The Zarankiewicz Problem for Polygon Visibility Gr…
Zarankiewicz's problem asks for the largest possible number of edges in a graph that does not contain a $K_{u,u}$ subgraph for a fixed positive integer $u$. Recently, Fox, Pach, Sheffer, Sulk and Zahl considered this problem for…
The `global' Zarankiewicz problem for hypergraphs asks for an upper bound on the number of edges of a finite $r$-hypergraph $V$ in terms of the number $|V|$ of its vertices, assuming the edge relation is induced by a fixed $K_{k, \dots,…
The classical Zarankiewicz's problem asks for the maximum number of edges in a bipartite graph on $n$ vertices which does not contain the complete bipartite graph $K_{t,t}$. In one of the cornerstones of extremal graph theory, K\H{o}v\'ari…
The classical Zarankiewicz problem, which concerns the maximum number of edges in a bipartite graph without a forbidden complete bipartite subgraph, motivates a direct analogue for hypergraphs. Let $K_{s_1,\ldots, s_r}$ be the complete…
Let $F$ be a graph, $k \geq 2$ be an integer, and write $\mathrm{ex}_{ \chi \leq k } (n , F)$ for the maximum number of edges in an $n$-vertex graph that is $k$-partite and has no subgraph isomorphic to $F$. The function $\mathrm{ex}_{ \chi…
A bipartite graph $H = \left(V_1, V_2; E \right)$ with $|V_1| + |V_2| = n$ is semilinear if $V_i \subseteq \mathbb{R}^{d_i}$ for some $d_i$ and the edge relation $E$ consists of the pairs of points $(x_1, x_2) \in V_1 \times V_2$ satisfying…
Given positive integers m,n,s,t, let z(m,n,s,t) be the maximum number of ones in a (0,1) matrix of size m-by-n that does not contain an all ones submatrix of size s-by-t. We find a flexible upper bound on z(m,n,s,t) that implies the known…
In this survey paper, we present open problems and conjectures on visibility graphs of points, segments and polygons along with necessary backgrounds for understanding them.
For positive integers $s,t,m$ and $n$, the Zarankiewicz number $z(m,n;s,t)$ is the maximum number of edges in a subgraph of $K_{m,n}$ that has no complete bipartite subgraph containing $s$ vertices in the part of size $m$ and $t$ vertices…
We improve Larman's bound on the diameter of a polytope by showing that if $\Delta$ is a normal simplicial complex, all of whose missing faces have size at most $r$, then the diameter of the facet-ridge graph of $\Delta$ is not larger than…
The hypergraph Zarankiewicz's problem, introduced by Erd\H{o}s in 1964, asks for the maximum number of hyperedges in an $r$-partite hypergraph with $n$ vertices in each part that does not contain a copy of $K_{t,t,\ldots,t}$. Erd\H{o}s…
The problem of Zarankiewicz asks for the maximum number of edges in a bipartite graph on $n$ vertices which does not contain the complete bipartite graph $K_{k,k}$ as a subgraph. A classical theorem due to K\H{o}v\'ari, S\'os, and Tur\'an…
A topological graph is $k$-quasi-planar if it does not contain $k$ pairwise crossing edges. A 20-year-old conjecture asserts that for every fixed $k$, the maximum number of edges in a $k$-quasi-planar graph on $n$ vertices is $O(n)$. Fox…
A bipartite graph $G$ is semi-algebraic in $\mathbb{R}^d$ if its vertices are represented by point sets $P,Q \subset \mathbb{R}^d$ and its edges are defined as pairs of points $(p,q) \in P\times Q$ that satisfy a Boolean combination of a…
For a bipartite graph $H$, its linear threshold is the smallest real number $\sigma$ such that every bipartite graph $G = (U \sqcup V, E)$ with unbalanced parts $|V| \gtrsim |U|^\sigma$ and without a copy of $H$ must have a linear number of…
A $t$-bar visibility representation of a graph assigns each vertex up to $t$ horizontal bars in the plane so that two vertices are adjacent if and only if some bar for one vertex can see some bar for the other via an unobstructed vertical…
The Zarankiewicz problem, a cornerstone problem in extremal graph theory, asks for the maximum number of edges in an $n$-vertex graph that does not contain the complete bipartite graph $K_{s,s}$. While the problem remains widely open in the…
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,…
The visibility graph of a finite set of points in the plane has the points as vertices and an edge between two vertices if the line segment between them contains no other points. This paper establishes bounds on the edge- and…
We prove upper bounds on the graph diameters of polytopes in two settings. The first is a worst-case bound for polytopes defined by integer constraints in terms of the height of the integers and certain subdeterminants of the constraint…