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 $\mathrm{Ex}(P,n)\geq n(\log n)^{\omega(1)}$.

翻译：禁止0-1矩阵的极值理论研究函数$\mathrm{Ex}(P,n)$的渐近增长，该函数定义为所有避免固定模式$P\in\{0,1\}^{k\times l}$的矩阵$A\in\{0,1\}^{n\times n}$的最大权重。该理论在解决组合学、离散与计算几何、结构图论以及数据结构分析（特别涉及动态最优性猜想的推论）中的问题方面取得了巨大成功。所有这些应用均采用无环模式，即当$P$被视为二部图的邻接矩阵时，该图为无环图。该领域最大的未解决问题是对无环$P$的$\mathrm{Ex}(P,n)$进行上界估计。先前的结果仅排除了Furedi和Hajnal猜想的严格$O(n\log n)$界。现有结果表明，既有可能对于所有$P$满足$\mathrm{Ex}(P,n)\leq n\log^{1+o(1)} n$，也有可能对于任意$\epsilon>0$存在$P$使得$\mathrm{Ex}(P,n) \geq n^{2-\epsilon}$。本文对无环$P$的极值函数建立了更强的下界：我们提出一种新构造方法，得到包含$\Theta(n(\log n/\log\log n)^t)$个1的相对稠密0-1矩阵，且该矩阵避免无环模式$X_t$。Pach和Tardos猜想此类结果已达最优，即不存在无环$P$使得$\mathrm{Ex}(P,n)\geq n(\log n)^{\omega(1)}$。