A Ramsey theorem for multiposets

Draganić, Nemanja; Mašulović, Dragan

doi:10.1016/j.ejc.2019.05.001

Cited by 4 publications

(5 citation statements)

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“…Similarly, superposing the linear order with the universal homogeneous structure in language with one unary relation it follows that the homogeneous dense local order has big Ramsey degrees (as shown by Laflamme, Nguyen Van Thé and Sauer [LNVTS10]). Superposing multiple linear and partial orders leads to big Ramsey equivalents of results of Sokić [Sok13], Solecki and Zhao [SZ17], and Draganić and Mašulović [DM19]. 7.…”

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confidence: 98%

Big Ramsey degrees using parameter spaces

Hubička¹

2020

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We show that the universal homogeneous partial order has finite big Ramsey degrees and discuss several corollaries. Our proof uses parameter spaces and the Carlson-Simpson theorem rather than (a strengthening of) the Halpern-Läuchli theorem and the Milliken tree theorem, which are the primary tools used to give bounds on big Ramsey degrees elsewhere (originating from work of Laver and Milliken). This new technique has many additional applications. To demonstrate this, we show that the homogeneous universal triangle-free graph has finite big Ramsey degrees, thus giving a short proof of a recent result of Dobrinen.

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confidence: 98%

Big Ramsey degrees using parameter spaces

Hubička¹

2020

Preprint

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“…, n} both P T and − → P T are Fraïssé ages. It was shown in [3] that the class − → P fin T has the Ramsey property, while Corollary 4.2 establishes the ordering property for the class. The rest is now an immediate consequence of Theorem 4.4.…”

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confidence: 93%

“…, n}. It was shown in [3] that the class − → P fin T has the Ramsey property, so Theorem 4.1 implies that it suffices to show that the class − → P T has the (W △C). But this is straightforward since for any Σ ⊆ S 2 ( − → P fin T ) we can always take τ = ({0, 1}, =, .…”

Section: If (Cmentioning

confidence: 99%

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The Ramsey and the Ordering Property for Classes of Lattices and Semilattices

Mašulović

2018

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The class of finite distributive lattices, as many other classes of structures, does not have the Ramsey property. It is quite common, though, that after expanding the structures with appropriately chosen linear orders the resulting class has the Ramsey property. So, one might expect that a similar result holds for the class of all finite distributive lattices. Surprisingly, Kechris and Sokić have proved in 2012 that this is not the case: no expansion of the class of finite distributive lattices by linear orders satisfies the Ramsey property.In this paper we prove that the variety of distributive lattices is not an exception, but an instance of a more general phenomenon. We show that for almost all nontrivial locally finite varieties of lattices no "reasonable" expansion of the finite members of the variety by linear orders gives rise to a Ramsey class. The responsibility for this lies not with the lattices as structures, but with the lack of algebraic morphisms: if we consider lattices as partially ordered sets (and thus switch from algebraic embeddings to embeddings of relational structures) we show that every variety of lattices gives rise to a class of linearly ordered posets having both the Ramsey property and the ordering property. It now comes as no surprise that the same is true for varieties of semilattices.

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“…Almost any reasonable class of finite relational structures has the Ramsey property or a Ramsey expansion, usually by an appropriate choice of linear orders. For example, finite graphs expanded with arbitrary linear orders have the Ramsey property [31,32]; the same holds for finite hypergraphs [31,32] and finite metric spaces [30]; finite posets expanded with linear orders that extend the partial order have the Ramsey property [39,11]; the same holds for multiposets -structures with several partial orders forming a partial order [9]; finite equivalence relations with linear orders where equivalence classes are convex have the Ramsey property [18]; the same holds for finite ultrametric spaces with linear orders where balls are convex [34,35]; and the list goes on and on. One of the most prominent general results in this direction is the Nešetřil-Rödl Theorem:…”

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confidence: 99%

Dual Ramsey properties for classes of algebras

Mašulović¹

2021

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Almost any reasonable class of finite relational structures has the Ramsey property or a Ramsey expansion. In contrast to that, the list of classes of finite algebras with the Ramsey expansion is surprisingly short. In this paper we show that any nontrivial variety (that is, equationally defined class of algebras) enjoys various dual Ramsey properties. We develop a completely new set of strategies that rely on the fact that right adjoints preserve the Ramsey property while left adjoints preserve the dual Ramsey property, and then treat classes of algebras as Eilenberg-Moore categories for a (co)monad. We first show that for any group G (finite or infinite) finite G-sets have finite small Ramsey degrees, and that every finite G-set has a finite big Ramsey degree in the cofree G-set on countably many cofree generators. We then show that finite algebras in any nontrivial variety have finite dual small Ramsey degrees, and that every finite algebra has finite dual big Ramsey degree in the free algebra on countably many free generators. As usual, these come as consequences of ordered versions of the statements. To the best of our knowledge, this is the first calculation of dual big Ramsey degrees after the Infinite Dual Ramsey Theorem of Carlson and Simpson.

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A Ramsey theorem for multiposets

Cited by 4 publications

References 23 publications

Big Ramsey degrees using parameter spaces

Big Ramsey degrees using parameter spaces

The Ramsey and the Ordering Property for Classes of Lattices and Semilattices

Dual Ramsey properties for classes of algebras

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