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 $\mathrm{Ex}(S_0,n),\mathrm{Ex}(S_1,n) \geq n2^{\Omega(\sqrt{\log n})}$, 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 $t\geq 2$, $\mathrm{Ex}(P_t,n)=\Theta(n(\log n/\log\log n)^t)$. This is the first proof of an asymptotically sharp bound that is $\omega(n\log n)$.

翻译：禁止0-1矩阵理论推广了Turán型（二分）子图回避问题、Davenport-Schinzel理论以及Zarankiewicz型问题，并在诸多领域产生重要影响，例如离散与计算几何、自调整数据结构的分析，以及图参数孪生宽度的理论发展。该领域最突出的开放问题是解决2005年提出的Pach-Tardos猜想，该猜想断言：若禁止模式$P\in\{0,1\}^{k\times l}$是无环图（森林）的二分关联矩阵，则$\mathrm{Ex}(P,n) = O(n\log^{C_P} n)$，其中$C_P$是仅依赖于$P$的常数。该猜想已在许多小型模式上得到证实，特别是所有权重不超过5的模式，以及除两个之外所有权重为6的模式。本文的主要成果是对Pach-Tardos猜想的清晰反驳。具体而言，我们证明$\mathrm{Ex}(S_0,n),\mathrm{Ex}(S_1,n) \geq n2^{\Omega(\sqrt{\log n})}$，其中$S_0,S_1$是尚未解决的两个权重为6的模式。我们还证明了整个交错模式族$(P_t)$的精确界，即对于所有$t\geq 2$，$\mathrm{Ex}(P_t,n)=\Theta(n(\log n/\log\log n)^t)$。这是首个渐近精确界达到$\omega(n\log n)$的证明。