A Refutation of the Pach-Tardos Conjecture for 0-1 Matrices
Seth Pettie, Gábor Tardos
TL;DR
The paper addresses the extremal theory of forbidden 0--1 matrices, focusing on acyclic patterns and the Pach–Tardos conjecture. It introduces a dense, Behrend-type construction and a signature-based framework to derive tight lower and upper bounds, culminating in a refutation of the Pach–Tardos conjecture by proving $\mathrm{Ex}(S_0,n),\mathrm{Ex}(S_1,n) \ge n2^{\sqrt{\log n}-O(\log\log n)}$ and establishing sharp bounds $\mathrm{Ex}(P_t,n)=\Theta\bigl(n(\log n/\log\log n)^t\bigr)$ for all $t\ge2$, thereby producing the first asymptotically sharp bound that grows faster than $n\log n$. The results extend to covering patterns and a hierarchy of $S_0^{(t)}$ patterns, showing that similar density amplification can be achieved under broader pattern-avoidance constraints. Moreover, these findings interact with related conjectures in ordered hypergraphs and edge-ordered graphs, refuting several conjectures in the extended Pach–Tardos framework. Overall, the work significantly advances understanding of the boundary between linear and polylogarithmic/exponential extremal growth in pattern-avoiding matrices and provides new tools and constructions for future investigations in combinatorics and related areas.
Abstract
The theory of forbidden 0-1 matrices generalizes Turan-style (bipartite) subgraph avoidance, Davenport-Schinzel theory, and Zarankiewicz-type problems, and has been influential in many areas, such as discrete and computational geometry, the analysis of self-adjusting data structures, and the development of the graph parameter twin width. The foremost open problems in this area is to resolve the Pach-Tardos conjecture from 2005, which states that if a forbidden pattern $P\in\{0,1\}^{k\times l}$ is the bipartite incidence matrix of an acyclic graph (forest), then $\mathrm{Ex}(P,n) = O(n\log^{C_P} n)$, where $C_P$ is a constant depending only on $P$. This conjecture has been confirmed on many small patterns, specifically all $P$ with weight at most 5, and all but two with weight 6. The main result of this paper is a clean refutation of the Pach-Tardos conjecture. Specifically, we prove that $\mathrm{Ex}(S_0,n),\mathrm{Ex}(S_1,n) \geq n2^{Ω(\sqrt{\log n})}$, where $S_0,S_1$ are the outstanding weight-6 patterns. We also prove sharp bounds on the entire class of alternating patterns $(P_t)$, specifically that for every $t\geq 2$, $\mathrm{Ex}(P_t,n)=Θ(n(\log n/\log\log n)^t)$. This is the first proof of an asymptotically sharp bound that is $ω(n\log n)$.