## Compressing and summarizing. Part 4.

[Disclaimer: this a blog-post consciously written in the style adumbrated in Compressing and summarizing. Part 0. For the original, see https://arxiv.org/abs/1708.00148. Compression ratio here is high. ]

This reports on this recent preprint by Siddharth Baskar and Alex Kruckman.

• Loc. cit. proves that for any purely relational signature $\Sigma$, and for any class $\mathbb{K}$ consisting of finite $\Sigma$-structures,

$\mathbb{K}$ has the strict order property

if and only if

$\mathbb{K}$ uniformly interprets the natural numbers.

• I recall that a logical formula $\psi$ having the strict order property relative to a theory $\mathbb{T}$ means that for every model $\mathfrak{A}$ of $\psi$, the poset $P := \{ (a_i,a_j)| \models\exists a\in \lvert \mathfrak{A}\rvert,\, ( \neg\psi(a,a_i)\wedge\psi(a,a_j) ) \}$ contains arbitrarily long finite chains.

(to be continued)