Related papers: A Refutation of the Pach-Tardos Conjecture for 0-1…

The extremal theory of forbidden 0-1 matrices studies the asymptotic growth of the function $\mathrm{Ex}(P,n)$, which is the maximum weight of a matrix $A\in\{0,1\}^{n\times n}$ whose submatrices avoid a fixed pattern $P\in\{0,1\}^{k\times…

For an $n$-vertex graph $G$, let $z(G;k)$ denote the number of zero forcing sets of size $k$. A conjecture of Boyer et al. asserts that the path $P_n$ maximizes these numbers coefficientwise among all $n$-vertex graphs; equivalently, the…

The Erd\H{o}s-Hajnal conjecture is one of the most classical and well-known problems in extremal and structural combinatorics dating back to 1977. It asserts that in stark contrast to the case of a general $n$-vertex graph if one imposes…

We confirm the eventual evasiveness of several classes of monotone graph properties under widely accepted number theoretic hypotheses. In particular we show that Chowla's conjecture on Dirichlet primes implies that (a) for any graph $H$,…

In 2006, Marcus and Tardos proved that if $A^1,\dots,A^n$ are cyclic orders on some subsets of a set of $n$ symbols such that the common elements of any two distinct orders $A^i$ and $A^j$ appear in reversed cyclic order in $A^i$ and $A^j$,…

Keszegh (2009) proved that the extremal function $ex(n, P)$ of any forbidden light $2$-dimensional 0-1 matrix $P$ is at most quasilinear in $n$, using a reduction to generalized Davenport-Schinzel sequences. We extend this result to…

Saturation problems for forbidden graphs have been a popular area of research for many decades, and recently Brualdi and Cao initiated the study of a saturation problem for 0-1 matrices. We say that 0-1 matrix $A$ is saturating for the…

We consider the following Tur\'an-type problem: given a fixed tournament $H$, what is the least integer $t=t(n,H)$ so that adding $t$ edges to any $n$-vertex tournament, results in a digraph containing a copy of $H$. Similarly, what is the…

In this thesis we consider ordered graphs (that is, graphs with a fixed linear ordering on their vertices). We summarize and further investigations on the number of edges an ordered graph may have while avoiding a fixed forbidden ordered…

A zero-one matrix $M$ is said to contain another zero-one matrix $A$ if we can delete some rows and columns of $M$ and replace some $1$-entries with $0$-entries such that the resulting matrix is $A$. The extremal number of $A$, denoted…

Given a fixed graph H, what is the (exponentially small) probability that the number X_H of copies of H in the binomial random graph G_{n,p} is at least twice its mean? Studied intensively since the mid 1990s, this so-called infamous upper…

Let $G$ be a graph with adjacency matrix $A(G)$. We conjecture that \[2n^+(G) \le n^-(G)(n^-(G) + 1),\] where $n^+(G)$ and $n^-(G)$ denote the number of positive and negative eigenvalues of $A(G)$, respectively. This conjecture generalizes…

Let $ex(n, P)$ be the maximum possible number of ones in any 0-1 matrix of dimensions $n \times n$ that avoids $P$. Matrix $P$ is called minimally non-linear if $ex(n, P) = \omega(n)$ but $ex(n, P') = O(n)$ for every strict subpattern $P'$…

Hajos' conjecture that every simple even graph on $n$ vertices can be decomposed into at most $(n-1)/2$ cycles (see L. Lovasz, On covering of graphs, in: P. Erdos, G.O.H. Katona (Eds.), Theory of Graphs, Academic Press, New York, 1968, pp.…

A conjecture of Erd\H{o}s from 1967 asserts that any graph on $n$ vertices which does not contain a fixed $r$-degenerate bipartite graph $F$ has at most $Cn^{2-1/r}$ edges, where $C$ is a constant depending only on $F$. We show that this…

An ordered graph $H$ is a simple graph with a linear order on its vertex set. The corresponding Tur\'an problem, first studied by Pach and Tardos, asks for the maximum number $\text{ex}_<(n,H)$ of edges in an ordered graph on $n$ vertices…

For a forbidden graph $L$, let $ex(p;L)$ denote the maximal number of edges in a simple graph of order $p$ not containing $L$. Let $T_n$ denote the unique tree on $n$ vertices with maximal degree $n-2$, and let $T_n^*=(V,E)$ be the tree on…

We present two extensions of the linear bound, due to Marcus and Tardos, on the number of 1's in an n by n 0-1 matrix avoiding a fixed permutation matrix. We first extend the linear bound to hypergraphs with ordered vertex sets and, using…

An efficient implicit representation of an $n$-vertex graph $G$ in a family $\mathcal{F}$ of graphs assigns to each vertex of $G$ a binary code of length $O(\log n)$ so that the adjacency between every pair of vertices can be determined…

We say a zero-one matrix $A$ avoids another zero-one matrix $P$ if no submatrix of $A$ can be transformed to $P$ by changing some ones to zeros. A fundamental problem is to study the extremal function $ex(n,P)$, the maximum number of…