Problem In Ramsey Theory Research Articles

Two problems in graph Ramsey theory

We study two problems in graph Ramsey theory. In the early 1970s, Erdős and O’Neil considered a generalization of Ramsey numbers. Given integers n,k,s and t with n≥k≥s,t≥2, they asked for the least integer N=fk(n,s,t) such that in any red–blue coloring of the k-subsets of {1,2,…,N}, there is a set of size n such that either each of its s-subsets is contained in some red k-subset, or each of its t-subsets is contained in some blue k-subset. Erdős and O’Neil found an exact formula for fk(n,s,t) when k≥s+t−1. In the arguably more interesting case where k=s+t−2, they showed 2−k2n<logfk(n,s,t)<2n for sufficiently large n. Our main result closes the gap between these lower and upper bounds, determining the logarithm of fs+t−2(n,s,t) up to a multiplicative factor.Recently, Damásdi, Keszegh, Malec, Tompkins, Wang and Zamora initiated the investigation of saturation problems in Ramsey theory, wherein one seeks to minimize n such that there exists an r-edge-coloring of Kn for which any extension of this to an r-edge-coloring of Kn+1 would create a new monochromatic copy of Kk. We obtain essentially sharp bounds for this problem.

On Ramsey-Minimal Infinite Graphs

For fixed finite graphs $G$, $H$, a common problem in Ramsey theory is to study graphs $F$ such that $F \to (G,H)$, i.e. every red-blue coloring of the edges of $F$ produces either a red $G$ or a blue $H$. We generalize this study to infinite graphs $G$, $H$; in particular, we want to determine if there is a minimal such $F$. This problem has strong connections to the study of self-embeddable graphs: infinite graphs which properly contain a copy of themselves. We prove some compactness results relating this problem to the finite case, then give some general conditions for a pair $(G,H)$ to have a Ramsey-minimal graph. We use these to prove, for example, that if $G=S_\infty$ is an infinite star and $H=nK_2$, $n \geqslant 1$ is a matching, then the pair $(S_\infty,nK_2)$ admits no Ramsey-minimal graphs.

The length of an s-increasing sequence of r-tuples

AbstractWe prove a number of results related to a problem of Po-Shen Loh [9], which is equivalent to a problem in Ramsey theory. Let a = (a1, a2, a3) and b = (b1, b2, b3) be two triples of integers. Define a to be 2-less than b if ai &lt; bi for at least two values of i, and define a sequence a1, …, am of triples to be 2-increasing if ar is 2-less than as whenever r &lt; s. Loh asks how long a 2-increasing sequence can be if all the triples take values in {1, 2, …, n}, and gives a log* improvement over the trivial upper bound of n2 by using the triangle removal lemma. In the other direction, a simple construction gives a lower bound of n3/2. We look at this problem and a collection of generalizations, improving some of the known bounds, pointing out connections to other well-known problems in extremal combinatorics, and asking a number of further questions.

Open Problems Column Edited by William Gasarch This Issue's Column!

In this column we state a class of open problems in Ramsey Theory. The general theme is to take Ramsey-type statements that are false and weaken them by allowing the homogenous set to use more than one color. This concept is not new, and the theorems we state and/or prove are not new; however, the open questions that request easier proofs of the known theorems (or weaker versions) may be new. We use the phrase an elementary proof. This is not meant to be a technical or rigorous term. What we really mean is a proof that can be taught in an undergraduate combinatorics course. A good example of what we mean is the proof of Theorem 9.3.

Exact Ramsey numbers of odd cycles via nonlinear optimisation

For a graph G, the k-colour Ramsey number Rk(G) is the least integer N such that every k-colouring of the edges of the complete graph KN contains a monochromatic copy of G. Let Cn denote the cycle on n vertices. We show that for fixed k≥2 and n odd and sufficiently large,Rk(Cn)=2k−1(n−1)+1. This resolves a conjecture of Bondy and Erdős for large n. The proof is analytic in nature, the first step of which is to use the regularity method to relate this problem in Ramsey theory to one in nonlinear optimisation. This allows us to prove a stability-type generalisation of the above and establish a correspondence between extremal k-colourings for this problem and perfect matchings in the k-dimensional hypercube Qk.

Down the large rabbit hole

This article documents my journey down the rabbit hole, chasing what I have come to know as a particularly unyielding problem in Ramsey theory on the integers: the 2-large conjecture. This conjecture states that if D⊆ℤ+ has the property that every 2-coloring of ℤ+ admits arbitrarily long monochromatic arithmetic progressions with common difference from D, then the same property holds for any finite number of colors. We hope to provide a roadmap for future researchers and also provide some new results related to the 2-large conjecture.

Constructions in Ramsey theory

We provide several constructions for problems in Ramsey theory. First, we prove a superexponential lower bound for the classical 4-uniform Ramsey number r 4 ( 5 , n ) , and the same for the iterated ( k − 4 ) -fold logarithm of the k-uniform version r k ( k + 1 , n ) . This is the first improvement of the original exponential lower bound for r 4 ( 5 , n ) implicit in work of Erdős and Hajnal from 1972 and also improves the current best known bounds for larger k due to the authors. Second, we prove an upper bound for the hypergraph Erdős–Rogers function f k + 1 , k + 2 k ( N ) that is an iterated ( k − 13 ) -fold logarithm in N. This improves the previous upper bounds that were only logarithmic and addresses a question of Dudek and the first author that was reiterated by Conlon, Fox and Sudakov. Third, we generalize the results of Erdős and Hajnal about the 3-uniform Ramsey number of K 4 minus an edge versus a clique to k-uniform hypergraphs.

The Multicolour Ramsey Number of a Long Odd Cycle

For a graph G, the k-colour Ramsey number Rk(G) is the least integer N such that every k-colouring of the edges of the complete graph KN contains a monochromatic copy of G. Bondy and Erdős conjectured that for an odd cycle Cn on n>3 vertices,Rk(Cn)=2k−1(n−1)+1. This is known to hold when k=2 and n>3, and when k=3 and n is large. We show that this conjecture holds asymptotically for k≥4, proving that for n odd,Rk(Cn)=2k−1n+o(n)asn→∞. The proof uses the regularity lemma to relate this problem in Ramsey theory to one in convex optimisation, allowing analytic methods to be exploited. Our analysis leads us to a new class of lower bound constructions for this problem, which naturally arise from perfect matchings in the k-dimensional hypercube. Progress towards a resolution of the conjecture for large n is also discussed.

Ramsey–Turán theory

Ramsey- and Turán-type problems were always strongly related to each other. Motivated by an observation of Paul Erdős, it was Turán who started the systematic investigation of the applications of extremal graph theory in geometry and analysis. This led the second author to some results and problems which, in turn, led to the birth of Ramsey–Turán-type theorems. Today this is a wide field of research with many interesting results and many unsolved problems. Below we give a short survey of the most important parts of this field: starting with a historical sketch we continue by describing the 1. Ramsey–Turán-type problems and results. 2. Related problems in Ramsey theory. 3. Some applications.

4566498 Reed control mechanism for terry loom: Gilmer A Williams, Douglas Burgess

For a graph G, the k-colour Ramsey number Rk(G) is the least integer N such that every k-colouring of the edges of the complete graph KN contains a monochromatic copy of G. Let Cn denote the cycle on n vertices. We show that for fixed k≥2 and n odd and sufficiently large, This resolves a conjecture of Bondy and Erdős for large n. The proof is analytic in nature, the first step of which is to use the regularity method to relate this problem in Ramsey theory to one in nonlinear optimisation. This allows us to prove a stability-type generalisation of the above and establish a correspondence between extremal k-colourings for this problem and perfect matchings in the k-dimensional hypercube Qk.