Universal graphs with forbidden subgraphs and algebraic closure

All those Ramsey classes (Ramsey classes with closures and forbidden homomorphisms)

We prove the Ramsey property for classes of ordered structures with closures and given local properties. This generalises earlier results: the Nešetřil-Rödl Theorem, the Ramsey property of partial orders and metric spaces as well as the authors' Ramsey lift of bowtie-free graphs. We use this framework to solve several open problems and give new examples of Ramsey classes. Among others, we find Ramsey lifts of convexly ordered $S$-metric spaces and prove the Ramsey theorem for finite models (i.e. structures with both functions and relations) thus providing the ultimate generalisation of the str…

Universal graphs with a forbidden subtree

We show that the problem of the existence of universal graphs with specified forbidden subgraphs can be systematically reduced to certain critical cases by a simple pruning technique which simplifies the underlying structure of the forbidden graphs, viewed as trees of blocks. As an application, we characterize the trees T for which a universal countable T-free graph exists.

有界灌木深度的禁止诱导子图与 MSO 的表达能力

Forbidden Induced Subgraphs for Bounded Shrub-Depth and the Expressive Power of MSO

通过禁止诱导子图刻画有界灌木深度图类，证明无界灌木深度的遗传类上 MSO 强于 FO，并解决 FO 与 MSO 表达力一致的开放问题。