Structural Ramsey Theory and the Extension Property for Partial Automorphisms

Hubička, Jan

doi:10.48550/arxiv.2010.05041

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2021

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“…Relative to the small Ramsey degree case (see e.g. [NVT15,Bod15,Hub20b]), there are fewer classes of structures for which big Ramsey degrees are fully understood. The following is the current state of the art:…”

Section: Introductionmentioning

confidence: 99%

Big Ramsey Degrees of the Generic Partial Order

Balko

Chodounský

Dobrinen

et al. 2021

Trends in Mathematics

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As a result of 33 intercontinental Zoom calls, we characterise big Ramsey degrees of the generic partial order. This is an infinitary extension of the well known fact that finite partial orders endowed with linear extensions form a Ramsey class (this result was announced by Nešetřil and Rödl in 1984 with first published proof by Paoli, Trotter and Walker in 1985). Towards this, we refine earlier upper bounds obtained by Hubička based on a new connection of big Ramsey degrees to the Carlson-Simpson theorem and we also introduce a new technique of giving lower bounds using an iterated application of the upper-bound theorem.

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Section: Introductionmentioning

confidence: 99%