Canonical Ramsey Theorem Research Articles

Ramsey Problems for Monotone Paths in Graphs and Hypergraphs

The study of ordered Ramsey numbers of monotone paths for graphs and hypergraphs has a long history, going back to the celebrated work by Erdős and Szekeres in the early days of Ramsey theory. In this paper we obtain several results in this area, establishing two conjectures of Mubayi and Suk and improving bounds due to Balko, Cibulka, Král and Kynčl. For example, in the graph case, we show that the ordered Ramsey number for a fixed clique versus a fixed power of a monotone path of length n is always linear in n. Also, in the 3-graph case, we show that the ordered Ramsey number for a fixed clique versus a tight monotone path of length n is always polynomial in n. As a by-product, we also obtain a color-monotone version of the well-known Canonical Ramsey Theorem of Erdős and Rado, which could be of independent interest.

On the Canonical Ramsey Theorem of Erdős and Rado and Ramsey Ultrafilters

On the Canonical Ramsey Theorem of Erdős and Rado and Ramsey Ultrafilters

A canonical Ramsey theorem with list constraints in random graphs

The celebrated canonical Ramsey theorem of Erdős and Rado implies that for a given graph H, if n is sufficiently large then any colouring of the edges of Kn gives rise to copies of H that exhibit certain colour patterns, namely monochromatic, rainbow or lexicographic. We are interested in sparse random versions of this result and the threshold at which the random graph G(n,p) inherits the canonical Ramsey properties of Kn. Our main result here pins down this threshold when we focus on colourings that are constrained by some prefixed lists. This result is applied in an accompanying work of the authors on the threshold for the canonical Ramsey property (with no list constraints) in the case that H is an even cycle.

The Number of Triple Systems Without Even Cycles

For k ⩾ 4, a loose k-cycle Ck is a hypergraph with distinct edges e1, e2, …, ek such that consecutive edges (modulo k) intersect in exactly one vertex and all other pairs of edges are disjoint. Our main result is that for every even integer k ⩾ 4, there exists c > 0 such that the number of triple systems with vertex set [n] containing no Ck is at most $$2^{cn^2}$$ . An easy construction shows that the exponent is sharp in order of magnitude. Our proof method is different than that used for most recent results of a similar flavor about enumerating discrete structures, since it does not use hypergraph containers. One novel ingredient is the use of some (new) quantitative estimates for an asymmetric version of the bipartite canonical Ramsey theorem.

Canonizing structural Ramsey theorems

At the beginning of 1950s Erdős and Rado suggested the investigation of the Ramsey-type results where the number of colors is not finite. This marked the birth of the so-called canonizing Ramsey theory. In 1985 Prömel and Voigt made the first step towards the structural canonizing Ramsey theory when they proved the canonical Ramsey property for the class of finite linearly ordered hypergraphs, and the subclasses thereof defined by forbidden substructures. Building on their results in this paper we provide several new structural canonical Ramsey results. We prove the canonical Ramsey theorem for the class of all finite linearly ordered tournaments, the class of all finite posets with linear extensions and the class of all finite linearly ordered metric spaces. We conclude the paper with the canonical version of the celebrated Nešetřil–Rödl Theorem. In contrast to the “classical” Ramsey-theoretic approach, in this paper we advocate the use of category theory to manage the complexity of otherwise technically overwhelming proofs typical in canonical Ramsey theory.

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.

An easier proof of the Canonical Ramsey Theorem

An easier proof of the Canonical Ramsey Theorem

Boolean algebras and Lubell functions

Let 2[n] denote the power set of [n], where [n]={1,2,…,n}. A collection B⊂2[n] forms a d-dimensional Boolean algebra if there exist pairwise disjoint sets X0,X1,…,Xd⊆[n], all non-empty with perhaps the exception of X0, so that B={X0∪⋃i∈IXi:I⊆[d]}. Let b(n,d) be the maximum cardinality of a family F⊂2X that does not contain a d-dimensional Boolean algebra. Gunderson, Rödl, and Sidorenko proved that b(n,d)≤cdn−1/2d⋅2n where cd=10d2−21−ddd−2−d.In this paper, we use the Lubell function as a new measurement for large families instead of cardinality. The Lubell value of a family of sets F with F⊆2[n] is defined by hn(F)=∑F∈F1/(n|F|). We prove the following Turán type theorem. If F⊆2[n] contains no d-dimensional Boolean algebra, then hn(F)≤2(n+1)1−21−d for sufficiently large n. This result implies b(n,d)≤Cn−1/2d⋅2n, where C is an absolute constant independent of n and d. With some modification, the ideas in Gunderson, Rödl, and Sidorenko's proof can be used to obtain this result. We apply the new bound on b(n,d) to improve several Ramsey-type bounds on Boolean algebras. We also prove a canonical Ramsey theorem for Boolean algebras.

A Canonical Ramsey Theorem for Exactly m-Coloured Complete Subgraphs

Given an edge colouring of a graph with a set of m colours, we say that the graph is exactly m-coloured if each of the colours is used. We consider edge colourings of the complete graph on $\mathbb{N}$ with infinitely many colours and show that either one can find an exactly m-coloured complete subgraph for every natural number m or there exists an infinite subset X ⊂ $\mathbb{N}$ coloured in one of two canonical ways: either the colouring is injective on X or there exists a distinguished vertex v in X such that X\{v} is 1-coloured and each edge between v and X\{v} has a distinct colour (all different to the colour used on X\{v}). This answers a question posed by Stacey and Weidl in 1999. The techniques that we develop also enable us to resolve some further questions about finding exactly m-coloured complete subgraphs in colourings with finitely many colours.

Applications of the Erdős-Rado Canonical Ramsey Theorem to Erdős-Type Problems

Let {p1,…,pn}⊆Rd. We think of d≪n. How big is the largest subset X of points such that all of the distances determined by elements of (X2) are different? We show that X is at least Ω((n1/(6d)(logn)1/3)/d1/3). This is not the best known; however the technique is new. Assume that no 3 of the original points are in the line. How big is the largest subset X of points such that all of the areas determined by elements of (X3) are different? If d=2 then the the size is at least Ω((loglogn)1/186); if d=3 then the size is at least Ω((loglogn)1/396). All of our proofs use variants of the canonical Ramsey theorem and some geometric lemmas.

Properly Edge-Coloured Subgraphs in Colourings of Bounded Degree

The smallest n such that every colouring of the edges of K n must contain a monochromatic star K 1,s+1 or a properly edge-coloured K t is denoted by f (s, t). Its existence is guaranteed by the Erdős–Rado Canonical Ramsey theorem and its value for large t was discussed by Alon, Jiang, Miller and Pritikin (Random Struct. Algorithms 23:409–433, 2003). In this note we primarily consider small values of t. We give the exact value of f (s, 3) for all s ≥ 1 and the exact value of f (2, 4), as well as reducing the known upper bounds for f (s, 4) and f (s, t) in general.

Constrained Ramsey Numbers

For two graphs S and T, the constrained Ramsey number f(S, T) is the minimum n such that every edge colouring of the complete graph on n vertices (with any number of colours) has a monochromatic subgraph isomorphic to S or a rainbow subgraph isomorphic to T. Here, a subgraph is said to be rainbow if all of its edges have different colours. It is an immediate consequence of the Erdős–Rado Canonical Ramsey Theorem that f(S, T) exists if and only if S is a star or T is acyclic. Much work has been done to determine the rate of growth of f(S, T) for various types of parameters. When S and T are both trees having s and t edges respectively, Jamison, Jiang and Ling showed that f(S, T) ≤ O(st2) and conjectured that it is always at most O(st). They also mentioned that one of the most interesting open special cases is when T is a path. In this paper, we study this case and show that f(S, Pt) = O(st log t), which differs only by a logarithmic factor from the conjecture. This substantially improves the previous bounds for most values of s and t.

The canonical Ramsey theorem and computability theory

Using the tools of computability theory and reverse mathematics, we study the complexity of two partition theorems, the Canonical Ramsey Theorem of Erdös and Rado, and the Regressive Function Theorem of Kanamori and McAloon. Our main aim is to analyze the complexity of the solutions to computable instances of these problems in terms of the Turing degrees and the arithmetical hierarchy. We succeed in giving a sharp characterization for the Canonical Ramsey Theorem for exponent 2 and for the Regressive Function Theorem for all exponents. These results rely heavily on a new, purely inductive, proof of the Canonical Ramsey Theorem. This study also unearths some interesting relationships between these two partition theorems, Ramsey’s Theorem, and König’s Lemma.

Constrained Ramsey Numbers

Abstract For two graphs S and T, the constrained Ramsey number f ( S , T ) is the minimum n such that every edge coloring of the complete graph on n vertices, with any number of colors, has a monochromatic subgraph isomorphic to S or a rainbow (all edges differently colored) subgraph isomorphic to T. The Erdős-Rado Canonical Ramsey Theorem implies that f ( S , T ) exists if and only if S is a star or T is acyclic, and much work has been done to determine the rate of growth of f ( S , T ) for various types of parameters. When S and T are both trees having s and t edges respectively, Jamison, Jiang, and Ling showed that f ( S , T ) ≤ O ( s t 2 ) and conjectured that it is always at most O ( s t ) . They also mentioned that one of the most interesting open special cases is when T is a path. In this work, we study this case and show that f ( S , P t ) = O ( s t log t ) , which differs only by a logarithmic factor from the conjecture. This substantially improves the previous bounds for most values of s and t.

Canonical Pattern Ramsey Numbers

A color pattern is a graph whose edges have been partitioned into color classes. A family [InlineMediaObject not available: see fulltext.] of color patterns is a Ramsey family provided there is some sufficiently large integer N such that in any edge coloring of the complete graph K N there is an (isomorphic) copy of at least one of the patterns from [InlineMediaObject not available: see fulltext.]. The smallest such N is the Ramsey number of the family [InlineMediaObject not available: see fulltext.]. The classical Canonical Ramsey theorem of Erd?s and Rado asserts that the family of color patterns is a Ramsey family if it consists of monochromatic, rainbow (totally multicolored) and lexically colored complete graphs. In this paper we treat the asymmetric case by studying the Ramsey number of families containing a rainbow triangle, a lexically colored complete graph and a fixed arbitrary monochromatic graph. In particular we give asymptotically tight bounds for the Ramsey number of a family consisting of rainbow and monochromatic triangle and a lexically colored K N . Among others, we prove some canonical Ramsey results for cycles.

On Pattern Ramsey Numbers of Graphs

A color pattern is a graph whose edges are partitioned into color classes. A family F of color patterns is a Ramsey family if there is some integer N such that every edge-coloring of KN has a copy of some pattern in F. The smallest such N is the (pattern) Ramsey number R(F) of F. The classical Canonical Ramsey Theorem of Erdos and Rado [4] yields an easy characterization of the Ramsey families of color patterns. In this paper we determine R(F) for all families consisting of equipartitioned stars, and we prove that ** when F consists of a monochromatic star of size s and a polychromatic triangle.

Essentially Infinite Colourings of Graphs

The classical canonical Ramsey theorem of Erdős and Rado states that, for any integer q ⩾ 1, any edge colouring of a large enough complete graph contains one of three canonically coloured complete subgraphs of order q. Of these canonical subgraphs, one is coloured monochromatically while each of the other two has its edge set coloured with many different colours. The paper presents a condition on colourings that, roughly speaking, requires them to make effective use of many colours (‘essential infiniteness’); this condition is then shown to imply that the colourings in question must contain large refinements of one of two ‘unavoidable’ colourings that are rich in colours. As it turns out, one of these unavoidable colourings is one of the canonical colourings of Erdős and Rado, and the other is a ‘bipartite variant’ of this colouring.

A Canonical Ramsey Theorem

Say a graph H selects a graph G if given any coloring of H, there will be a monochromatic induced copy of G in H or a completely multicolored copy of G in H. Denote by s(G) the minimum order of a graph that selects G and set s(n) = max {s(G): |G| = n}. Upper and lower bounds are given for this function. Also, consider the Folkman function fr(n) = max{min{|V(H)|: H → (G)1r}: |V(G)| = n}, where H → (G)1r indicates that H is vertex Ramsey to G, that is, any vertex coloring of H with r colors admits a monochromatic induced copy of G. The method used provides a better upper bound for this function than was previously known. As a tool, we establish a theorem for projective planes.