On linear forbidden submatrices

Abstract

In this paper we study the extremal problem of finding how many 1 entries an n by n 0-1 matrix can have if it does not contain certain forbidden patterns as submatrices. We call the number of 1 entries of a 0-1 matrix its weight. The extremal function of a pattern is the maximum weight of an n by n 0-1 matrix that does not contain this pattern as a submatrix. We call a pattern (a 0-1 matrix) linear if its extremal function is O (n). Our main results are modest steps towards the elusive goal of characterizing linear patterns. We find novel ways to generate new linear patterns from known ones and use this to prove the linearity of some patterns. We also find the first minimal non-linear pattern of weight above 4. We also propose an infinite sequence of patterns that we conjecture to be minimal non-linear but have Ω(n log n) as their extremal function. We prove a weaker statement only, namely that there are infinitely many minimal not quasi-linear patterns among the submatrices of these matrices. For the definition of these terms see below.

On Nonlinear Forbidden 0–1 Matrices: A Refutation of a Füredi-Hajnal Conjecture

“…In this section we give tight or nearly tight bounds on some low weight matrices and reprove a result due to Keszegh [17] and Geneson [15] that there are infinitely many minimal nonlinear matrices with respect to containment and stretching. Like their proof, ours is nonconstructive.…”

“…Although the forbidden pattern is probably not of any particular interest, our method for constructing matrices of density Θ(log n log log n) uses two generic composition procedures on 0-1 matrices that might be of interest to anyone exploring the landscape of forbidden 0-1 matrices or their applications. In Section 3 we give a substantially simpler proof [17,15] that there are infinitely many minimal nonlinear patterns with respect to containment. Our technique lets us prove that patterns in Keszegh's class [17] are nonlinear, as well as several previously unclassified ones.…”

“…In Section 3 we give a substantially simpler proof [17,15] that there are infinitely many minimal nonlinear patterns with respect to containment. Our technique lets us prove that patterns in Keszegh's class [17] are nonlinear, as well as several previously unclassified ones.…”

“…Following [11,4], Füredi and Hajnal [14], and Tardos [33] managed to categorize the growth of Ex(P, n) for every weight-4 pattern P and Tardos [33] bounded the growth of Ex(P, n) for most sets P of weight-4 patterns. However, our current understanding of weight-5 and larger forbidden patterns is incomplete [33,28,10,17,15]. Moreover, we have no characterization of patterns with linear, near-linear, or polynomial extremal functions.…”

“…At about the same time Bienstock and Györi [4] bounded the running time of Mitchell's algorithm [25], which finds shortest paths avoiding n-vertex obstacles in the plane, as a function of Ex(P 2 , n) = Θ(n log n) [4,33]. In subsequent Figure 1: Following a common convention [33,17,15] we write forbidden 0-1 matrices with bullets for 1s and blanks for 0s.…”