We show that Ramsey theory, a domain presently conceived to guarantee the existence of large homogeneous sets for partitions on k-tuples of words (for every natural number k) over a finite alphabet, can be extended to one for partitions on Schreier-type sets of words (of every countable ordinal). Indeed, we establish an extension of the partition theorem of Carlson about words and of the (more general) partition theorem of Furstenberg-Katznelson about combinatorial subspaces of the set of words (generating from k-tuples of words for any fixed natural number k) into a partition theorem about combinatorial subspaces (generating from Schreier-type sets of words of order any fixed countable ordinal). Furthermore, as a result we obtain a strengthening of Carlson's infinitary Nash-Williams type (and Ellentuck type) partition theorem about infinite sequences of variable words into a theorem, in which an infinite sequence of variable words and a binary partition of all the finite sequences of words, one of whose components is, in addition, a tree, are assumed, concluding that all the Schreier-type finite reductions of an infinite reduction of the given sequence have a behavior determined by the Cantor-Bendixson ordinal index of the tree-component of the partition, falling in the tree-component above that index and in its complement below it.
We prove a linear recurrence relation for a large family of generalized Schreier sets, which generalizes the Fibonacci recurrence proved by Bird and higher order Fibonacci recurrence proved by the second author et al. Furthermore, we show a relationship between Schreier-type sets and Turán graphs.
Ramsey theory for words over a finite alphabet was unified in the work of Carlson and Furstenberg-Katznelson. Carlson, in the same work, outlined a method to extend the theory for words over an infinite alphabet, but subject to a fixed dominating principle, proving in particular an Ellentuck version, and a corresponding Ramsey theorem for k=1. In the present work we develop in a systematic way a Ramsey theory for words (in fact for ω-Z*-located words) over a doubly infinite alphabet extending Carlson's approach (to countable ordinals and Schreier-type families), and we apply this theory, explo…
Counting Schreier Sets Under Neighborhood Conditions
UC San Diego（加州大学圣迭戈分校）；Washington and Lee University（华盛顿与李大学）；University of Akron（阿克伦大学）；University of Nottingham（诺丁汉大学）；Williams College（威廉姆斯学院）；University of Chicago（芝加哥大学）；University of Cincinnati Blue Ash（辛辛那提大学布卢什校区）
Recently, a relation between Schreier-type sets and Turán graphs was discovered. In this note, we give a combinatorial proof and obtain a generalization of the relation. Specifically, for $p, q\ge 1$, let $$\mathcal{A}_q := \{F\subset\mathbb{N}: |F| = 1 \mbox{ or }F\mbox{ is an arithmetic progression with difference } q\}$$ and $$Sr(n, p, q)\ :=\ \#\{F\subset \{1, \ldots, n\}\,:\, p\min F\ge |F|\mbox{ and }F\in \mathcal{A}_q\}.$$ We show that $$Sr(n, p, q) \ =\ T(n+1, pq+1, q),$$ where $T(\cdot, \cdot, \cdot)$ is the number of edges of an $n$-vertex graph that is a modification of Turán graphs…
A set $A$ of positive integers is said to be Schreier if either $A = \emptyset$ or $\min A\ge |A|$. We give a bijective map to prove the recurrence of the sequence $(|\mathcal{K}_{n, p, q}|)_{n=1}^\infty$ (for fixed $p\ge 1$ and $q\ge 2$), where $$\mathcal{K}_{n, p, q} \ = \ \{A\subset \{1, \ldots, n\}\,:\, \mbox{either }A = \emptyset \mbox{ or } (\max A-\max_2 A = p\mbox{ and }\min A\ge |A|\ge q)\}$$ and $\max_2 A$ is the second largest integer in $A$, given that $|A|\ge 2$. When $p = 1$ and $q=2$, we have that $(|\mathcal{K}_{n, 1, 2}|)_{n=1}^\infty$ is the Fibonacci sequence. As a corollary…
We present a refinement of Ramsey numbers by considering graphs with a partial ordering on their vertices. This is a natural extension of the ordered Ramsey numbers. We formalize situations in which we can use arbitrary families of partially-ordered sets to form host graphs for Ramsey problems. We explore connections to well studied Turán-type problems in partially-ordered sets, particularly those in the Boolean lattice. We find a strong difference between Ramsey numbers on the Boolean lattice and ordered Ramsey numbers when the partial ordering on the graphs have large antichains.
A finite set $A \subset \mathbb{R}^d$ is called $\textit{diameter-Ramsey}$ if for every $r \in \mathbb N$, there exists some $n \in \mathbb N$ and a finite set $B \subset \mathbb{R}^n$ with $\mathrm{diam}(A)=\mathrm{diam}(B)$ such that whenever $B$ is coloured with $r$ colours, there is a monochromatic set $A' \subset B$ which is congruent to $A$. We prove that sets of diameter $1$ with circumradius larger than $1/\sqrt{2}$ are not diameter-Ramsey. In particular, we obtain that triangles with an angle larger than $135^\circ$ are not diameter-Ramsey, improving a result of Frankl, Pach, Reiher a…
A finite set $X$ in some Euclidean space $R^n$ is called Ramsey if for any $k$ there is a $d$ such that whenever $R^d$ is $k$-coloured it contains a monochromatic set congruent to $X$. This notion was introduced by Erdos, Graham, Montgomery, Rothschild, Spencer and Straus, who asked if a set is Ramsey if and only if it is spherical, meaning that it lies on the surface of a sphere. This question (made into a conjecture by Graham) has dominated subsequent work in Euclidean Ramsey theory. In this paper we introduce a new conjecture regarding which sets are Ramsey; this is the first ever `rival' c…
We characterize a class of topological Ramsey spaces such that each element $\mathcal R$ of the class induces a collection $\{\mathcal R_k\}_{k<ω}$ of projected spaces which have the property that every Baire set is Ramsey. Every projected space $\mathcal R_k$ is a subspace of the corresponding space of length-$k$ approximation sequences with the Tychonoff, equivalently metric, topology. This answers a question of S. Todorcevic and generalizes the results of Carlson \cite{Carlson}, Carlson-Simpson \cite{CarSim2}, Prömel-Voigt \cite{PromVoi}, and Voigt \cite{Voigt}. We also present a new family…
We determine several generalised Ramsey numbers for two sets $Γ_1$ and $Γ_2$ of cycles, in particular, all generalised Ramsey numbers $R(Γ_1,Γ_2)$ such that $Γ_1$ or $Γ_2$ contains a cycle of length at most $6$, or the shortest cycle in each set is even. This generalises previous results of Erdős, Faudree, Rosta, Rousseau, and Schelp from the 1970s. Notably, including both $C_3$ and $C_4$ in one of the sets, makes very little difference from including only $C_4$. Furthermore, we give a conjecture for the general case. We also describe many $(Γ_1,Γ_2)$-avoiding graphs, including a complete char…
Let $k$ and $p$ be positive integers and let $Q$ be a finite point set in general position in the plane. We say that $Q$ is $(k,p)$-Ramsey if there is a finite point set $P$ such that for every $k$-coloring $c$ of $\binom{P}{p}$ there is a subset $Q'$ of $P$ such that $Q'$ and $Q$ have the same order type and $\binom{Q'}{p}$ is monochromatic in $c$. Nešetřil and Valtr proved that for every $k \in \mathbb{N}$, all point sets are $(k,1)$-Ramsey. They also proved that for every $k \ge 2$ and $p \ge 2$, there are point sets that are not $(k,p)$-Ramsey. As our main result, we introduce a new family…
In this paper, we investigate a variant of Ramsey numbers called defective Ramsey numbers where cliques and independent sets are generalized to $k$-dense and $k$-sparse sets, both commonly called $k$-defective sets. We focus on the computation of defective Ramsey numbers restricted to some subclasses of perfect graphs. Since direct proof techniques are often insufficient for obtaining new values of defective Ramsey numbers, we provide a generic algorithm to compute defective Ramsey numbers in a given target graph class. We combine direct proof techniques with our efficient graph generation alg…
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 probabil…
We say that a 0-1 matrix $N$ of size $a\times b$ can be found in a collection of sets $\mathcal{H}$ if we can find sets $H_{1}, H_{2}, \dots, H_{a}$ in $\mathcal{H}$ and elements $e_1, e_2, \dots, e_b$ in $\cup_{H \in \mathcal{H}} H$ such that $N$ is the incidence matrix of the sets $H_{1}, H_{2}, \dots, H_{a}$ over the elements $e_1, e_2, \dots, e_b$. We prove the following Ramsey-type result: for every $n\in \N$, there exists a number S(n) such that in any collection of at least S(n) sets, one can find either the incidence matrix of a collection of $n$ singletons, or its complementary matrix…
A set of $N$ permutations of $\{1,2,\ldots,v\}$ is $t$-suitable, if each symbol precedes each subset of $t-1$ others in at least one permutation. The extremal problem of determining the smallest size $N$ of such sets for given $v$ and $t$ was the subject of classical studies by Dushnik in 1950 and Spencer in 1971. Colbourn recently introduced the concept of suitable cores as equivalent objects of suitable sets of permutations, and studied the dual problem of determining the largest $v=\text{SCN}(t,N)$ such that a suitable core exists for given $t$ and $N$. Chan and Jedwab showed that when $N=\…
In this note we consider a Ramsey type result for partially ordered sets. In particular, we give an alternative short proof of a theorem for a posets with multiple linear extensions recently obtained by Solecki and Zhao.
The Theorems of Hindman and van der Waerden belong to the classical theorems of partition Ramsey Theory. The Central Sets Theorem is a strong simultaneous extension of both theorems that applies to general commutative semigroups. We give a common extension of the Central Sets Theorem and Ramsey's Theorem.
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.
Many important problems in combinatorics and other related areas can be phrased in the language of independent sets in hypergraphs. Recently Balogh, Morris and Samotij, and independently Saxton and Thomason developed very general container theorems for independent sets in hypergraphs; both of which have seen numerous applications to a wide range of problems. In this paper we use the container method to give relatively short and elementary proofs of a number of results concerning Ramsey (and Turán) properties of (hyper)graphs and the integers. In particular: (i) We generalise the random Ramsey …
In 1964, Erdős, Hajnal and Moon introduced a saturation version of Turán's classical theorem in extremal graph theory. In particular, they determined the minimum number of edges in a $K_r$-free, $n$-vertex graph with the property that the addition of any further edge yields a copy of $K_r$. We consider analogues of this problem in other settings. We prove a saturation version of the Erdős-Szekeres theorem about monotone subsequences and saturation versions of some Ramsey-type theorems on graphs and Dilworth-type theorems on posets. We also consider semisaturation problems, wherein we allow the…
We explore the properties of non-piecewise syndetic sets with positive upper density, which we call "discordant", in countably infinite amenable (semi)groups. Sets of this kind are involved in many questions of Ramsey theory and manifest the difference in complexity between the classical van der Waerden's theorem and Szemerédi's theorem. We generalize and unify old constructions and obtain new results about these historically interesting sets. Along the way, we draw from various corners of mathematics, including classical Ramsey theory, ergodic theory, number theory, and topological and symbol…
Artin-Schreier geproci configurations in projective spaces of arbitrary dimension
Dipartimento di Ingegneria dell’Informazione e Scienze Matematiche, Università di Siena（锡耶纳大学）；Department of Mathematics, University of the National Education Commission（国家教育委员会大学）；Dipartimento di Ingegneria, Università degli Studi di Palermo（巴勒莫大学）；Department of Mathematics, University of Nebraska–Lincoln（内布拉斯加大学林肯分校）；Department of Mathematics, University of Notre Dame（圣母大学）
It is shown that Matet's characterization of the Ramsey property relative to a selective co-ideal $\mathcal{H}$, in terms of games of Kastanas, still holds if we consider semiselectivity instead of selectivity. Moreover, we prove that a co-ideal $\mathcal{H}$ is semiselective if and only if Matet's game-theoretic characterization of the $\mathcal{H}$-Ramsey property holds. This lifts Kastanas's characterization of the classical Ramsey property to its optimal setting, from the point of view of the local Ramsey theory and gives a game-theoretic counterpart to a theorem of Farah \cite{far}, asser…
We give a probabilistic construction of a $3$-uniform hypergraph on $N$ vertices with independence number $O(\log N / \log \log N)$ in which there are at most two edges among any four vertices. This bound is tight and solves a longstanding open problem of Erdős and Hajnal in Ramsey theory. We further extend this result to prove tight bounds on various other hypergraph Ramsey numbers.
In this note we establish a Ramsey-type result for certain subsets of the $n$-dimensional cube. This can then be applied to obtain reasonable bounds on various related structures, such as (partial) Hales-Jewett lines for alphabets of sized 3 and 4, Hilbert cubes in sets of real numbers with small sumsets, "corner" in the integer lattice in the plane, and 3-term geometric progressions in integers.
A k-ary semi-algebraic relation E on R^d is a subset of R^{kd}, the set of k-tuples of points in R^d, which is determined by a finite number of polynomial equations and inequalities in kd real variables. The description complexity of such a relation is at most t if the number of polynomials and their degrees are all bounded by t. A subset A of R^d is called homogeneous if all or none of the k-tuples from A satisfy E. A large number of geometric Ramsey-type problems and results can be formulated as questions about finding large homogeneous subsets of sets in R^d equipped with semi-algebraic rel…
In a recent work, N. Hindman, D. Strauss and L. Zamboni have shown that the Hales-Jewett theorem can be combined with a sufficiently well behaved homomorphisms. Their work was completely algebraic in nature, where they have used the algebra of Stone-Cech compactification of discrete semigroup. They have proved the existence of those configurations in piecewise syndetic sets, which is a Ramsey theoretic rich set. In our work we will show those forms are still present in very small but Ramsey theoretic sets, (like J-set, C-set) and our proof is purely elementary in nature.
In this paper we extend the block combinatorics partition theorems of Hindman and Milliken in the setting of the recursive system of the block Schreier families (B^xi) consisting of families defined for every countable ordinal xi. Results contain (a) a block partition Ramsey theorem for every countable ordinal xi (Hindman's theorem corresponding to xi=1, and Milliken's theorem to xi a finite ordinal), (b) a countable ordinal form of the block Nash-Williams partition theorem, and (c) a countable ordinal block partition theorem for sets closed in the infinite block analogue of Ellentuck's topolo…
We study the structure of red-blue edge colorings of complete graphs, with no copies of the $n$-cycle $C_n$ in red, and no copies of the $n$-wheel $W_n = C_n \ast K_1$ in blue, for an odd integer $n$. Our first main result is that in any such coloring, deleting at most two vertices we obtain a vertex-partition of $G$ into three sets such that the edges inside the partition classes are red, and edges between partition classes are blue. As a second result, we obtain bounds for the Ramsey numbers of $r(C_{2k+1},W_{2j})$ for $k < j$ integers, which asymptotically confirm the values of $4j+1$, as i…
We generalise the results by Bigorajska and Kotlarski about partitioning $α$-large sets, by extending the domain up to ordinals below $\varepsilon_ω$. These results will be very useful to give a miniaturisation of the infinite Ramsey Theorem.
A question of Erdős asks if for every pair of positive integers $r$ and $k$, there exists a graph $H$ having $\textrm{girth}(H)=k$ and the property that every $r$-colouring of the edges of $H$ yields a monochromatic cycle $C_k$. The existence of such graphs was confirmed by the third author and Ruciński. We consider the related numerical problem of determining the smallest such graph with this property. We show that for integers $r$ and $k$, there exists a graph $H$ on $R^{10k^2} k^{15k^3}$ vertices (where $R = R(C_k;r)$ is the $r$-colour Ramsey number for the cycle $C_k$) having $\textrm{girt…
It is observed that the conjugacy growth series of the infinite finitary symmetric group with respect to the generating set of transpositions is the generating series of the partition function. Other conjugacy growth series are computed, for other generating sets, for restricted permutational wreath products of finite groups by the finitary symmetric group, and for alternating groups. Similar methods are used to compute usual growth polynomials and conjugacy growth polynomials for finite symmetric groups and alternating groups, with respect to various generating sets of transpositions. Computa…
For a fixed infinite graph $H$, we study the largest density of a monochromatic subgraph isomorphic to $H$ that can be found in every two-coloring of the edges of $K_{\mathbb{N}}$. This is called the Ramsey upper density of $H$, and was introduced by Erdős and Galvin. Recently, the Ramsey upper density of the infinite path was determined. Here, we find the value of this density for all locally finite graphs $H$ up to a factor of 2, answering a question of DeBiasio and McKenney. We also find the exact density for a wide class of bipartite graphs, including all locally finite forests. Our approa…
