arXiv subjects

Timothy Trujillo

Publications and source records attributed to Timothy Trujillo.

5 recordsLinked to original sources

arXiv2016

Triangular Ramsey Numbers

The purpose of this paper is to introduce the idea of triangular Ramsey numbers and provide values as well as upper and lower bounds for them. To do this, the combinatorial game Mines is introduced; after some necessary theorems about triangular sets are proved. This game is easy enough that young children are able to play. The most basic variations of this game are analyzed and theorems about winning strategies and the existence of draws are proved. The game of Mines is then used to define triangular Ramsey numbers. Lower bounds are found for these triangular Ramsey numbers using the probabilistic method and the theorems about triangular sets.

math.CO↗

arXiv2016

From abstract alpha-Ramsey theory to abstract ultra-Ramsey theory

We work within the framework of the Alpha-Theory introduced by Benci and Di Nasso. The Alpha-Theory postulates a few natural properties for an infinite "ideal" number $α$. The formulation provides an elementary axiomatics for the methods of abstract ultra-Ramsey theory. The main results are Theorem 10, Theorem 57, Theorem 67 and Theorem 73. Theorem 10 is an infinite-dimensional extension of the celebrated Ramsey's Theorem. We show that corollaries of this result include the Galvin-Pirky Theorem, the Silver Theorem and the $\vecα$-Ellentuck Theorem. We prove that, under the assumption of the $\mathfrak{c}^{+}$-enlarging property, the $\vecα$-Ellentuck Theorem is equivalent to the Ultra-Ellentuck Theorem of Todorcevic. Theorem 57 is an abstraction of Theorem 10 to the setting of triples $(\mathcal{R},\le,r)$ where $\mathcal{R}\not=\emptyset$, $\le$ is a quasi-order on $\mathcal{R}$ and $r$ is a function with domain $\mathbb{N}\times \mathcal{R}$. We use Theorem 57 to develop the Abstract $\vecα$-Ellentuck Theorem, Theorem 67, and the Abstract Ultra-Ellentuck Theorem, Theorem 73, extending the Abstract Ellentuck Theorem along the same lines as the $\vecα$-Ellentuck Theorem and Ultra-Ellentuck Theorem extend the Ellentuck Theorem, respectively. We conclude with some examples illustrating the theory and an open question related to the local Ramsey theory developed by Di Prisco, Mijares and Nieto.

math.LO↗

arXiv2015

Topological Ramsey spaces from Fraïssé classes, Ramsey-classification theorems, and initial structures in the Tukey types of p-points

A general method for constructing a new class of topological Ramsey spaces is presented. Members of such spaces are infinite sequences of products of Fraïssé classes of finite relational structures satisfying the Ramsey property. The Product Ramsey Theorem of Sokič is extended to equivalence relations for finite products of structures from Fraïssé classes of finite relational structures satisfying the Ramsey property and the Order-Prescribed Free Amalgamation Property. This is essential to proving Ramsey-classification theorems for equivalence relations on fronts, generalizing the Pudlák-Rödl Theorem to this class of topological Ramsey spaces. To each topological Ramsey space in this framework corresponds an associated ultrafilter satisfying some weak partition property. By using the correct Fraïssé classes, we construct topological Ramsey spaces which are dense in the partial orders of Baumgartner and Taylor in \cite{Baumgartner/Taylor78} generating p-points which are $k$-arrow but not $k+1$-arrow, and in a partial order of Blass in \cite{Blass73} producing a diamond shape in the Rudin-Keisler structure of p-points. Any space in our framework in which blocks are products of $n$ many structures produces ultrafilters with initial Tukey structure exactly the Boolean algebra $\mathcal{P}(n)$. If the number of Fraïssé classes on each block grows without bound, then the Tukey types of the p-points below the space's associated ultrafilter have the structure exactly $[ω]^{<ω}$. In contrast, the set of isomorphism types of any product of finitely many Fraïssé classes of finite relational structures satisfying the Ramsey property and the OPFAP, partially ordered by embedding, is realized as the initial Rudin-Keisler structure of some p-point generated by a space constructed from our template.

math.LO↗

arXiv2014

Ramsey for $\mathcal{R}_{1}$ ultrafilter mappings and their Dedekind cuts

Associated to each ultrafilter $\mathcal{U}$ on $ω$ and each map $p:ω\rightarrow ω$ is a Dedekind cut in the ultrapower $ω^ω/p( \mathcal{U})$. Blass has characterized, under CH, the cuts obtainable when $\mathcal{U}$ is taken to be either a p-point ultrafilter, a weakly-Ramsey ultrafilter or a Ramsey ultrafilter. Dobrinen and Todorcevic have introduced the topological Ramsey space $\mathcal{R}_{1}$. Associated to the space $\mathcal{R}_{1}$ is a notion of Ramsey ultrafilter for $\mathcal{R}_{1}$ generalizing the familiar notion of Ramsey ultrafilter on $ω$. We characterize, under CH, the cuts obtainable when $\mathcal{U}$ is taken to be a Ramsey for $\mathcal{R}_{1}$ ultrafilter and $p$ is taken to be any map. In particular, we show that the only cut obtainable is the standard cut, whose lower half consists of the collection of equivalence classes of constants maps. Forcing with $\mathcal{R}_{1}$ using almost-reduction adjoins an ultrafilter which is Ramsey for $\mathcal{R}_{1}$. For such ultrafilters $\mathcal{U}_{1}$, Dobrinen and Todorcevic have shown that the Rudin-Keisler types of the p-points within the Tukey type of $\mathcal{U}_{1}$ consists of a strictly increasing chain of rapid p-points of order type $ω$. We show that for any Rudin-Keisler mapping between any two p-points within the Tukey type of $\mathcal{U}_{1}$ the only cut obtainable is the standard cut. These results imply existence theorems for special kinds of ultrafilters.

math.LO↗

arXiv2013

Selective but not Ramsey

We give a partial answer to the following question of Dobrinen: For a given topological Ramsey space $\mathcal{R}$, are the notions of selective for $\mathcal{R}$ and Ramsey for $\mathcal{R}$ equivalent? Every topological Ramsey space $\mathcal{R}$ has an associated notion of Ramsey ultrafilter for $\mathcal{R}$ and selective ultrafilter for $\mathcal{R}$ (see \cite{MijaresSelective}). If $\mathcal{R}$ is taken to be the Ellentuck space then the two concepts reduce to the familiar notions of Ramsey and selective ultrafilters on $ω$; so by a well-known result of Kunen the two are equivalent. We give the first example of an ultrafilter on a topological Ramsey space that is selective but not Ramsey for the space, and in fact a countable collection of such examples. For each positive integer $n$ we show that for the topological Ramsey space $\mathcal{R}_{n}$ from \cite{Ramsey-Class2}, the notions of selective for $\mathcal{R}_{n}$ and Ramsey for $\mathcal{R}_{n}$ are not equivalent. In particular, we prove that forcing with a closely related space using almost-reduction, adjoins an ultrafilter that is selective but not Ramsey for $\mathcal{R}_{n}$. Moreover, we introduce a notion of finite product among members of the family $\{\mathcal{R}_{n}: n<ω\}$. We show that forcing with closely related product spaces using almost-reduction, adjoins ultrafilters that are selective but not Ramsey for these product topological Ramsey spaces.

math.LO↗