We establish a list of characterizations of bounded twin-width for hereditary, totally ordered binary structures. This has several consequences. First, it allows us to show that a (hereditary) class of matrices on a finite alphabet either contains at least $n!$ matrices of size $n \times n$, or at most $c^n$ for some constant $c$. This generalizes the celebrated Stanley-Wilf conjecture/Marcus-Tardos theorem from permutation classes to any matrix class on a finite alphabet, answers our small conjecture [SODA '21] in the case of ordered graphs, and with more work, settles a question first asked by Balogh, Bollob\'as, and Morris [Eur. J. Comb. '06] on the growth of hereditary classes of ordered graphs. Second, it gives a fixed-parameter approximation algorithm for twin-width on ordered graphs. Third, it yields a full classification of fixed-parameter tractable first-order model checking on hereditary classes of ordered binary structures. Fourth, it provides an alternative proof to a model-theoretic characterization of classes with bounded twin-width announced by Simon and Toru\'nczyk.

翻译：我们为世袭、完全有序的二进制结构建立一份双曲线捆绑式双曲线特征清单。 这有几种后果。 首先, 它让我们能够显示, 固定字母上的( 遗传) 矩阵( 遗传) 类别包含至少 $n\ timen n$, 或最多 $c$, 或最多 $c$ 。 这概括了 著名的 Stanley- Wilf 参数/ Marcus- Tardos 理论从固定字母表单到任何矩阵类, 在有定序图表的情况下, 回答我们的小假设 [SODA'21], 并且做更多的工作, 解决了Balogh、 Bollob\'as 和 Morris [Eur. J. Comb. '06] 首次询问的关于定序图遗传类增长的问题。 其次, 它为定购图上的双曲线上的双曲线提供了固定参数近比值算法。 第三, 它生成了固定参数直径第一级模型的完整分类, 检查定序双曲线结构结构结构结构结构的模型。 第四, 它向Simal- 提供替代证据。