Linear bounds on matrix extremal functions using visibility hypergraphs

The 0-1 matrix A contains a 0-1 matrix M if some submatrix of A can be transformed into M by changing some ones to zeroes. If A does not contain M, then A avoids M. Let ex(n,M) be the maximum number of ones in an n x n 0-1 matrix that avoids M, and let ex_k(m,M) be the maximum number of columns in a 0-1 matrix with m rows that avoids M and has at least k ones in every column. A method for bounding ex(n,M) by using bounds on the maximum number of edges in bar visibility graphs was introduced in (R. Fulek, Discrete Mathematics 309, 2009). By using a similar method with bar visibility hypergraphs, we obtain linear bounds on the extremal functions of other forbidden 0-1 matrices.Comment: 11 pages, 4 figure

Extremal problems in ordered graphs

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 graph as a subgraph. In particular, we take a step toward confirming a conjecture of Pach and Tardos regarding the number of edges allowed when the forbidden pattern is a tree by establishing an upper bound for a particular ordered graph for which existing techniques have failed. We also generalize a theorem of Geneson by establishing an upper bound on the number of edges allowed if the forbidden graphs fit a generalized notion of a matching.Comment: Thesis for Master Degree, Simon Fraser Universit

Ordered and convex geometric trees with linear extremal function

The extremal functions $ex_{\rightarrow}(n,F)$ and ex_{\cir}(n,F) for ordered and convex geometric acyclic graphs $F$ have been extensively investigated by a number of researchers. Basic questions are to determine when $ex_{\rightarrow}(n,F)$ and ex_{\cir}(n,F) are linear in $n$, the latter posed by Bra\ss-K\'arolyi-Valtr in 2003. In this paper, we answer both these questions for every tree $F$. We give a forbidden subgraph characterization for a family $\cal T$ of ordered trees with $k$ edges, and show that $ex_{\rightarrow}(n,T) = (k - 1)n - {k \choose 2}$ for all $n \geq k + 1$ when $T \in {\cal T}$ and $ex_{\rightarrow}(n,T) = \Omega(n\log n)$ for $T \not\in {\cal T}$. We also describe the family of the convex geometric trees with linear Tur\' an number and show that for every convex geometric tree $F$ not in this family, ex_{\cir}(n,F)= \Omega(n\log \log n).Comment: 14 pages, 9 figures. Same as the first version. Only metadata has been change

Sharp Bounds on Davenport-Schinzel Sequences of Every Order

One of the longest-standing open problems in computational geometry is to bound the lower envelope of $n$ univariate functions, each pair of which crosses at most $s$ times, for some fixed $s$. This problem is known to be equivalent to bounding the length of an order-$s$ Davenport-Schinzel sequence, namely a sequence over an $n$-letter alphabet that avoids alternating subsequences of the form $a \cdots b \cdots a \cdots b \cdots$ with length $s+2$. These sequences were introduced by Davenport and Schinzel in 1965 to model a certain problem in differential equations and have since been applied to bounding the running times of geometric algorithms, data structures, and the combinatorial complexity of geometric arrangements. Let $\lambda_s(n)$ be the maximum length of an order-$s$ DS sequence over $n$ letters. What is $\lambda_s$ asymptotically? This question has been answered satisfactorily (by Hart and Sharir, Agarwal, Sharir, and Shor, Klazar, and Nivasch) when $s$ is even or $s\le 3$. However, since the work of Agarwal, Sharir, and Shor in the mid-1980s there has been a persistent gap in our understanding of the odd orders. In this work we effectively close the problem by establishing sharp bounds on Davenport-Schinzel sequences of every order $s$. Our results reveal that, contrary to one's intuition, $\lambda_s(n)$ behaves essentially like $\lambda_{s-1}(n)$ when $s$ is odd. This refutes conjectures due to Alon et al. (2008) and Nivasch (2010).Comment: A 10-page extended abstract will appear in the Proceedings of the Symposium on Computational Geometry, 201

Forbidden induced subposets of given height

Let $P$ be a partially ordered set. The function \mbox{La}^{\#}(n,P) denotes the size of the largest family $\mathcal{F}\subset 2^{[n]}$ that does not contain an induced copy of $P$. It was proved by Methuku and P\'alv\"olgyi that there exists a constant $C_{P}$ (depending only on $P$) such that \mbox{La}^{\#}(n,P). However, the order of the constant $C_{P}$ following from their proof is typically exponential in $|P|$. Here, we show that if the height of the poset is constant, this can be improved. We show that for every positive integer $h$ there exists a constant $c_{h}$ such that if $P$ has height at most $h$, then \mbox{La}^{\#}(n,P)\leq |P|^{c_{h}}\binom{n}{\lfloor n/2\rfloor}. Our methods also immediately imply that similar bounds hold in grids as well. That is, we show that if $\mathcal{F}\subset [k]^{n}$ such that $\mathcal{F}$ does not contain an induced copy of $P$ and $n\geq 2|P|$, then $$|\mathcal{F}|\leq |P|^{c_{h}}w,$$ where $w$ is the width of $[k]^{n}$. A small part of our proof is to partition $2^{[n]}$ (or $[k]^{n}$) into certain fixed dimensional grids of large sides. We show that this special partition can be used to derive bounds in a number of other extremal set theoretical problems and their generalizations in grids, such as the size of families avoiding weak posets, Boolean algebras, or two distinct sets and their union. This might be of independent interest.Comment: 17 pages, 2 figure

Almost all permutation matrices have bounded saturation functions

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 forbidden 0-1 matrix $P$ if $A$ avoids $P$ but changing any zero to a one in $A$ creates a copy of $P$. Define $sat(n, P)$ to be the minimum possible number of ones in an $n \times n$ 0-1 matrix that is saturating for $P$. Fulek and Keszegh proved that for every 0-1 matrix $P$, either $sat(n, P) = O(1)$ or $sat(n, P) = \Theta(n)$. They found two 0-1 matrices $P$ for which $sat(n, P) = O(1)$, as well as infinite families of 0-1 matrices $P$ for which $sat(n, P) = \Theta(n)$. Their results imply that $sat(n, P) = \Theta(n)$ for almost all $k \times k$ 0-1 matrices $P$. Fulek and Keszegh conjectured that there are many more 0-1 matrices $P$ such that $sat(n, P) = O(1)$ besides the ones they found, and they asked for a characterization of all permutation matrices $P$ such that $sat(n, P) = O(1)$. We affirm their conjecture by proving that almost all $k \times k$ permutation matrices $P$ have $sat(n, P) = O(1)$. We also make progress on the characterization problem, since our proof of the main result exhibits a family of permutation matrices with bounded saturation functions

On the Extremal Functions of Acyclic Forbidden 0--1 Matrices

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 $P\in\{0,1\}^{k\times l}$. This theory has been wildly successful at resolving problems in combinatorics, discrete and computational geometry, structural graph theory, and the analysis of data structures, particularly corollaries of the dynamic optimality conjecture. All these applications use acyclic patterns, meaning that when $P$ is regarded as the adjacency matrix of a bipartite graph, the graph is acyclic. The biggest open problem in this area is to bound $\mathrm{Ex}(P,n)$ for acyclic $P$. Prior results have only ruled out the strict $O(n\log n)$ bound conjectured by Furedi and Hajnal. It is consistent with prior results that $\forall P. \mathrm{Ex}(P,n)\leq n\log^{1+o(1)} n$, and also consistent that $\forall \epsilon>0.\exists P. \mathrm{Ex}(P,n) \geq n^{2-\epsilon}$. In this paper we establish a stronger lower bound on the extremal functions of acyclic $P$. Specifically, we give a new construction of relatively dense 0--1 matrices with $\Theta(n(\log n/\log\log n)^t)$ 1s that avoid an acyclic $X_t$. Pach and Tardos have conjectured that this type of result is the best possible, i.e., no acyclic $P$ exists for which $\mathrm{Ex}(P,n)\geq n(\log n)^{\omega(1)}$

Formations and generalized Davenport-Schinzel sequences

An $(r, s)$-formation is a concatenation of $s$ permutations of $r$ distinct letters. We define the function $F_{r, s}(n)$ to be the maximum possible length of a sequence with $n$ distinct letters that avoids all $(r, s)$-formations and has every $r$ consecutive letters distinct, and we define the function $F_{r, s}(n, m)$ to be the maximum possible length of a sequence with $n$ distinct letters that avoids all $(r, s)$-formations and can be partitioned into $m$ blocks of distinct letters. (Nivasch, 2010) and (Pettie, 2015) found bounds on $F_{r, s}(n)$ for all fixed $r, s > 0$, but no exact values were known, even for $s = 2$. We prove that $F_{r,2}(n, m) = n+(r-1)(m-1)$, $F_{r,3}(n, m) = 2n+(r-1)(m-2)$, $F_{r,2}(n) = (n-r)r+2r-1$, and $F_{r,3}(n) = 2(n-r)r+3r-1$, improving on bounds of (Klazar, 1992), (Nivasch, 2010), and (Pettie, 2015). In addition, we improve an upper bound of (Klazar, 2002). For any sequence $u$, define $Ex(u, n)$ to be the maximum possible length of a sequence with $n$ distinct letters that avoids $u$ and has every $r$ consecutive letters distinct, where $r$ is the number of distinct letters in $u$. Klazar proved that $Ex((a_1 \dots a_r)^2, n) < (2n+1)L$, where $L = Ex((a_1 \dots a_r)^2,K-1)+1$ and $K = (r-1)^4 + 1$. Here we prove that $K = (r-1)^4 + 1$ in Klazar's bound can be replaced with $K = (r-1)^3+1$. We also prove a conjecture from (Geneson et al., 2014) by showing for $t \geq 1$ that $Ex(a b c (a c b)^{t} a b c, n) = n 2^{\frac{1}{t!}\alpha(n)^{t} \pm O(\alpha(n)^{t-1})}$. In addition, we prove that $Ex(a b c a c b (a b c)^{t} a c b, n) = n 2^{\frac{1}{(t+1)!}\alpha(n)^{t+1} \pm O(\alpha(n)^{t})}$ for $t \geq 1$. Furthermore, we extend the equalities $F_{r,2}(n, m) = n+(r-1)(m-1)$ and $F_{r,3}(n, m) = 2n+(r-1)(m-2)$ to formations in $d$-dimensional 0-1 matrices, sharpening a bound from (Geneson, 2019)

A generalization of the K\H{o}v\'{a}ri-S\'{o}s-Tur\'{a}n theorem

We present a new proof of the K\H{o}v\'{a}ri-S\'{o}s-Tur\'{a}n theorem that $ex(n, K_{s,t}) = O(n^{2-1/t})$ for $s, t \geq 2$. The new proof is elementary, avoiding the use of convexity. For any $d$-uniform hypergraph $H$, let $ex_d(n,H)$ be the maximum possible number of edges in an $H$-free $d$-uniform hypergraph on $n$ vertices. Let $K_{H, t}$ be the $(d+1)$-uniform hypergraph obtained from $H$ by adding $t$ new vertices $v_1, \dots, v_t$ and replacing every edge $e$ in $E(H)$ with $t$ edges $e \cup \left\{v_1\right\},\dots, e \cup \left\{v_t\right\}$ in $E(K_{H, t})$. If $H$ is the $1$-uniform hypergraph on $s$ vertices with $s$ edges, then $K_{H, t} = K_{s, t}$. We prove that $ex_{d+1}(n,K_{H,t}) = O(ex_d(n, H)^{1/t} n^{d+1-d/t} + t n^d)$ for any $d$-uniform hypergraph $H$ with at least two edges such that $ex_d(n, H) = o(n^d)$. Thus $ex_{d+1}(n,K_{H,t}) = O(n^{d+1-1/t})$ for any $d$-uniform hypergraph $H$ with at least two edges such that $ex_d(n, H) = O(n^{d-1})$, which implies the K\H{o}v\'{a}ri-S\'{o}s-Tur\'{a}n theorem in the $d = 1$ case. This also implies that $ex_{d+1}(n, K_{H,t}) = O(n^{d+1-1/t})$ when $H$ is a $d$-uniform hypergraph with at least two edges in which all edges are pairwise disjoint, which generalizes an upper bound proved by Mubayi and Verstra\"{e}te (JCTA, 2004). We also obtain analogous bounds for 0-1 matrix Tur\'{a}n problems

Sharper bounds and structural results for minimally nonlinear 0-1 matrices

The extremal function $ex(n, P)$ is the maximum possible number of ones in any 0-1 matrix with $n$ rows and $n$ columns that avoids $P$. A 0-1 matrix $P$ is called minimally non-linear if $ex(n, P) = \omega(n)$ but $ex(n, P') = O(n)$ for every $P'$ that is contained in $P$ but not equal to $P$. Bounds on the maximum number of ones and the maximum number of columns in a minimally non-linear 0-1 matrix with $k$ rows were found in (CrowdMath, 2018). In this paper, we improve the bound on the maximum number of ones in a minimally non-linear 0-1 matrix with $k$ rows from $5k-3$ to $4k-4$. As a corollary, this improves the upper bound on the number of columns in a minimally non-linear 0-1 matrix with $k$ rows from $4k-2$ to $4k-4$. We also prove that there are not more than four ones in the top and bottom rows of a minimally non-linear matrix and that there are not more than six ones in any other row of a minimally non-linear matrix. Furthermore, we prove that if a minimally non-linear 0-1 matrix has ones in the same row with exactly $d$ columns between them, then within these columns there are at most $2d-1$ rows above and $2d-1$ rows below with ones