Ramsey's Theorem Research Articles

Ramsey theorem for designs

We prove that for every choice of parameters 2≤t≤k and 1≤λ the class PD→ktλ of linearly ordered partial designs with parameters k, t, λ is a Ramsey class. Thus, together with the recent spectacular results of Keevash, one obtains that the class of linearly ordered designs D→ktλ is a Ramsey class.

The Complexity of Phylogeny Constraint Satisfaction Problems

We systematically study the computational complexity of a broad class of computational problems in phylogenetic reconstruction. The class contains, for example, the rooted triple consistency problem, forbidden subtree problems, the quartet consistency problem, and many other problems studied in the bioinformatics literature. The studied problems can be described as constraint satisfaction problems , where the constraints have a first-order definition over the rooted triple relation. We show that every such phylogeny problem can be solved in polynomial time or is NP-complete. On the algorithmic side, we generalize a well-known polynomial-time algorithm of Aho, Sagiv, Szymanski, and Ullman for the rooted triple consistency problem. Our algorithm repeatedly solves linear equation systems to construct a solution in polynomial time. We then show that every phylogeny problem that cannot be solved by our algorithm is NP-complete. Our classification establishes a dichotomy for a large class of infinite structures that we believe is of independent interest in universal algebra, model theory, and topology. The proof of our main result combines results and techniques from various research areas: a recent classification of the model-complete cores of the reducts of the homogeneous binary branching C-relation, Leeb’s Ramsey theorem for rooted trees, and universal algebra.

Forcing-theoretic aspects of Hindman's Theorem

We investigate the partial order $(\mathrm{FIN})^\omega$ of infinite block sequences, ordered by almost condensation, from the forcing-theoretic point of view. This order bears the same relationship to Hindman's Theorem as $\mathcal{P}(\omega) /\mathrm{fin}$ does to Ramsey's Theorem. While $(\mathcal{P}(\omega) / \mathrm{fin})^2$ completely embeds into $(\mathrm{FIN})^\omega$, we show this is consistently false for higher powers of $\mathcal{P} (\omega) / \mathrm{fin}$, by proving that the distributivity number $\mathfrak{h}_3$ of $(\mathcal{P} (\omega) /\mathrm{fin})^3$ may be strictly smaller than the distributivity number $\mathfrak{h}_{\mathrm{FIN}}$ of $(\mathrm{FIN})^\omega$. We also investigate infinite maximal antichains in $(\mathrm{FIN})^\omega$ and show that the least cardinality $\mathfrak{a}_{\mathrm{FIN}}$ of such a maximal antichain is at least the smallest size of a nonmeager set of reals. As a consequence, we obtain that $\mathfrak{a}_{\mathrm{FIN}}$ is consistently larger than $\mathfrak{a}$, the least cardinality of an infinite maximal antichain in $\mathcal{P} (\omega) / \mathrm{fin}$.

Detection of Emergent Situations in Complex Systems by Structural Invariant (MB, M)

The paper introduces complete description of the detection method that uses structural invariant Matroid and its Bases (MB, M). There are recapitulated essential concepts from the used knowledge field as “complex system, emergent situations (A, B, C)”, Ramsey theorem and principal computation variables “power” and “complexity” of emergence phenomenon. The method is explained in details and the demonstration of its application is done by the detection of emergent situation – violation of Short Water Cycle in an ecosystem.

A “quantum” Ramsey theorem for operator systems

Let V \mathcal {V} be a linear subspace of M n ( C ) M_n(\mathbb {C}) which contains the identity matrix and is stable under the formation of Hermitian adjoints. We prove that if n n is sufficiently large, then there exists a rank k k orthogonal projection P P such that dim ( P V P ) = 1 \textrm {dim}(P\mathcal {V}P) = 1 or k 2 k^2 .

An infinite self-dual Ramsey theorem

In a recent paper [5] S. Solecki proved a finite self-dual Ramsey theorem that extends simultaneously the classical finite Ramsey theorem [4] and the Graham–Rothschild theorem [2]. In this paper we prove the corresponding infinite dimensional version of the self-dual theorem. As a consequence, we extend the classical infinite Ramsey theorem [4] and the Carlson–Simpson theorem [1].

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 Fraisse classes of finite relational structures satisfying the Ramsey property. The Product Ramsey Theorem of Sokic is extended to equivalence relations for finite products of structures from Fraisse 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 Pudlak–Rodl 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 Fraisse classes, we construct topological Ramsey spaces which are dense in the partial orders of Baumgartner and Taylor (Trans Am Math Soc 241:283–309, 1978) generating p-points which are k-arrow but not $$k+1$$ -arrow, and in a partial order of Blass (Trans Am Math Soc 179:145–166, 1973) 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 Fraisse 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 $$[\omega ]^{<\omega }$$ . In contrast, the set of isomorphism types of any product of finitely many Fraisse 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.

Strong Ramsey games: Drawing on an infinite board

Consider the following strong Ramsey game. Two players take turns in claiming a previously unclaimed edge of the complete graph on n vertices until all edges have been claimed. The first player to build a copy of Kq (q≥3) is declared the winner of the game. If none of the players win, then the game ends in a draw. A simple strategy stealing argument shows that the second player cannot expect to ever win this game. Moreover, for sufficiently large n, it follows from Ramsey's Theorem that the game cannot end in a draw and is thus a first player win. A famous question of Beck asks whether the minimum number of moves needed for the first player to win this game on Kn grows with n. This seems unlikely but is still wide open. A striking equivalent formulation of this question is whether the same game played on the infinite complete graph is a first player win or a draw.The target graph of the strong Ramsey game does not have to be Kq: it can be any predetermined fixed graph. In fact, it can even be a k-uniform hypergraph (and then the game is played on the infinite k-uniform hypergraph). Since strategy stealing and Ramsey's Theorem still apply, one can ask the same question: is this game a first player win or a draw? The same intuition which led most people (including the authors) to believe that the Kq strong Ramsey game on the infinite board is a first player win, would also lead one to believe that the H strong Ramsey game on the infinite board is a first player win for any uniform hypergraph H. However, in this paper we construct a 5-uniform hypergraph for which the corresponding game is a draw.

The strength of infinitary Ramseyan principles can be accessed by their densities

In this article, we conduct a model-theoretic investigation of three infinitary Ramseyan statements: the Infinite Ramsey Theorem for pairs and two colours (RT22), the Canonical Ramsey Theorem for pairs (CRT2) and the Regressive Ramsey Theorem for pairs (RegRT2). We approximate the logical strength of these principles by the strength of their first-order iterated versions, known as density principles. We then investigate their logical strength using strong initial segments of models of Peano Arithmetic, in the spirit of the classical article by Paris and Kirby, hereby re-proving old results model-theoretically. The article is concluded by a discussion of two further outreaches of densities. One is a further investigation of the strength of the Ramsey Theorem for pairs. The other deals with the asymptotics of the standard Ramsey function R22.

Gowers' Ramsey theorem with multiple operations and dynamics of the homeomorphism group of the Lelek fan

We generalize the finite version of Gowers' Ramsey theorem to multiple tetris-like operations and apply it to show that a group of homeomorphisms that preserve a “typical” linear order of branches of the Lelek fan, a compact connected metric space with many symmetries, is extremely amenable.

Gowers' Ramsey Theorem for generalized tetris operations

We prove a generalization of Gowers' theorem for FINk where, instead of the single tetris operation T:FINk→FINk−1, one considers all maps from FINk to FINj for 0≤j≤k arising from nondecreasing surjections f:{0,1,…,k}→{0,1,…,j}. This answers a question of Bartošová and Kwiatkowska. We also describe how to prove a common generalization of such a result and the Galvin–Glazer–Hindman theorem on finite products, in the setting of layered partial semigroups introduced by Farah, Hindman, and McLeod.

Constructions and nonexistence results for suitable sets of permutations

A set of N permutations of {1,2,…,v} is (N,v,t)-suitable if each symbol precedes each subset of t−1 others in at least one permutation. The central problems are to determine the smallest N for which such a set exists for given v and t, and to determine the largest v for which such a set exists for given N and t. These extremal problems were the subject of classical studies by Dushnik in 1950 and Spencer in 1971. We give examples of suitable sets of permutations for new parameter triples (N,v,t). We relate certain suitable sets of permutations with parameter t to others with parameter t+1, thereby showing that one of the two infinite families recently presented by Colbourn can be constructed directly from the other. We prove an exact nonexistence result for suitable sets of permutations using elementary combinatorial arguments. We then establish an asymptotic nonexistence result using Ramsey's theorem.

The inductive strength of Ramsey's Theorem for Pairs

We address the question, “Which number theoretic statements can be proven by computational means and applications of Ramsey's Theorem for Pairs?” We show that, over the base theory RCA0, Ramsey's Theorem for Pairs does not imply Σ20-induction.

Controlling iterated jumps of solutions to combinatorial problems

Among the Ramsey-type hierarchies, namely, Ramsey’s theorem, the free set, the thin set and the rainbow Ramsey theorem, only Ramsey’s theorem is known to collapse in reverse mathematics. A promising approach to show the strictness of the hierarchies would be to prove that every computable instance at level n has a lown solution. In particular, this requires effective control of iterations of the Turing jump. In this paper, we design some variants of Mathias forcing to construct solutions to cohesiveness, the Erdős–Moser theorem and stable Ramsey’s theorem for pairs, while controlling their iterated jumps. For this, we define forcing relations which, unlike Mathias forcing, have the same definitional complexity as the formulas they force. This analysis enables us to answer two questions of Wei Wang, namely, whether cohesiveness and the Erdős–Moser theorem admit preservation of the arithmetic hierarchy, and can be seen as a step towards the resolution of the strictness of the Ramsey-type hierarchies.

Computable Ramsey’s theorem for pairs needs infinitely many $$\Pi ^0_2$$ Π 2 0 sets

In Ramsey's Theorem and Recursion Theory, Theorem 4.2, Jockusch proved that for any computable k-coloring of pairs of integers, there is an infinite $$\Pi ^0_2$$ź20 homogeneous set. The proof used a countable collection of $$\Pi ^0_2$$ź20 sets as potential infinite homogeneous sets. In a remark preceding the proof, Jockusch stated without proof that it can be shown that there is no computable way to prove this result with a finite number of $$\Pi ^0_2$$ź20 sets. We provide a proof of this claim, showing that there is no computable way to take an index for an arbitrary computable coloring and produce a finite number of indices of $$\Pi ^0_2$$ź20 sets with the property that one of those sets will be homogeneous for that coloring. While proving this result, we introduce n-trains as objects with useful combinatorial properties which can be used as approximations to infinite $$\Pi ^0_2$$ź20 sets.

Hindman’s Theorem is only a Countable Phenomenon

We pursue the idea of generalizing Hindman's Theorem to uncountable cardinalities, by analogy with the way in which Ramsey's Theorem can be generalized to weakly compact cardinals. But unlike Ramsey's Theorem, the outcome of this paper is that the natural generalizations of Hindman's Theorem proposed here tend to fail at all uncountable cardinals.

Dominating the Erdős–Moser theorem in reverse mathematics

The Erdős–Moser theorem (EM) states that every infinite tournament has an infinite transitive subtournament. This principle plays an important role in the understanding of the computational strength of Ramsey's theorem for pairs (RT22) by providing an alternate proof of RT22 in terms of EM and the ascending descending sequence principle (ADS). In this paper, we study the computational weakness of EM and construct a standard model (ω-model) of simultaneously EM, weak König's lemma and the cohesiveness principle, which is not a model of the atomic model theorem. This separation answers a question of Hirschfeldt, Shore and Slaman, and shows that the weakness of the Erdős–Moser theorem goes beyond the separation of EM from ADS proven by Lerman, Solomon and Towsner.

On the Dual Ramsey Property for Finite Distributive Lattices

The class of finite distributive lattices, as many other natural 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 Sokic 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 class of finite distributive lattices does not have the dual Ramsey property either. However, we are able to derive a dual Ramsey theorem for finite distributive lattices endowed with a particular linear order. Both results are consequences of the recently observed fact that categorical equivalence preserves the Ramsey property.

A Ramsey theorem for partial orders with linear extensions

We prove a Ramsey theorem for finite sets equipped with a partial order and a fixed number of linear orders extending the partial order. This is a common generalization of two recent Ramsey theorems due to Sokić. As a bonus, our proof gives new arguments for these two results.

On the Proof Complexity of Paris-Harrington and Off-Diagonal Ramsey Tautologies

We study the proof complexity of Paris-Harrington’s Large Ramsey Theorem for bi-colorings of graphs and of off-diagonal Ramsey’s Theorem. For Paris-Harrington, we prove a non-trivial conditional lower bound in Resolution and a non-trivial upper bound in bounded-depth Frege. The lower bound is conditional on a (very reasonable) hardness assumption for a weak (quasi-polynomial) Pigeonhole principle in R es (2). We show that under such an assumption, there is no refutation of the Paris-Harrington formulas of size quasi-polynomial in the number of propositional variables. The proof technique for the lower bound extends the idea of using a combinatorial principle to blow up a counterexample for another combinatorial principle beyond the threshold of inconsistency. A strong link with the proof complexity of an unbalanced off-diagonal Ramsey principle is established. This is obtained by adapting some constructions due to Erdős and Mills. We prove a non-trivial Resolution lower bound for a family of such off-diagonal Ramsey principles.