Ramsey Graphs Research Articles

Riemannian Manifolds, Closed Geodesic Lines, Topology and Ramsey Theory

We applied the Ramsey analysis to the sets of points belonging to Riemannian manifolds. The points are connected with two kinds of lines: geodesic and non-geodesic. This interconnection between the points is mapped into the bi-colored, complete Ramsey graph. The selected points correspond to the vertices of the graph, which are connected with the bi-colored links. The complete bi-colored graph containing six vertices inevitably contains at least one mono-colored triangle; hence, a mono-colored triangle, built of the green or red links, i.e., non-geodesic or geodesic lines, consequently appears in the graph. We also considered the bi-colored, complete Ramsey graphs emerging from the intersection of two Riemannian manifolds. Two Riemannian manifolds, namely (M1,g1) and (M2,g2), represented by the Riemann surfaces which intersect along the curve (M1,g1)∩(M2,g2)=ℒ were addressed. Curve ℒ does not contain geodesic lines in either of the manifolds (M1,g1) and (M2,g2). Consider six points located on the ℒ: {1,…6}⊂ℒ. The points {1,…6}⊂ℒ are connected with two distinguishable kinds of the geodesic lines, namely with the geodesic lines belonging to the Riemannian manifold (M1,g1)/red links, and, alternatively, with the geodesic lines belonging to the manifold (M2,g2)/green links. Points {1,…6}⊂ℒ form the vertices of the complete graph, connected with two kinds of links. The emerging graph contains at least one closed geodesic line. The extension of the theorem to the Riemann surfaces of various Euler characteristics is presented.

Converting Tessellations into Graphs: From Voronoi Tessellations to Complete Graphs

A mathematical procedure enabling the transformation of an arbitrary tessellation of a surface into a bi-colored, complete graph is introduced. Polygons constituting the tessellation are represented by vertices of the graphs. Vertices of the graphs are connected by two kinds of links/edges, namely, by a green link, when polygons have the same number of sides, and by a red link, when the polygons have a different number of sides. This procedure gives rise to a semi-transitive, complete, bi-colored Ramsey graph. The Ramsey semi-transitive number was established as Rtrans(3,3)=5 Shannon entropies of the tessellation and graphs are introduced. Ramsey graphs emerging from random Voronoi and Poisson Line tessellations were investigated. The limits ζ=limN→∞NgNr, where N is the total number of green and red seeds, Ng and Nr, were found ζ= 0.272 ± 0.001 (Voronoi) and ζ= 0.47 ± 0.02 (Poisson Line). The Shannon Entropy for the random Voronoi tessellation was calculated as S= 1.690 ± 0.001 and for the Poisson line tessellation as S = 1.265 ± 0.015. The main contribution of the paper is the calculation of the Shannon entropy of the random point process and the establishment of the new bi-colored Ramsey graph on top of the tessellations.

SAT Modulo Symmetries for Graph Generation and Enumeration

We propose a novel SAT-based approach to graph generation. Our approach utilizes the interaction between a CDCL SAT solver and a special symmetry propagator where the SAT solver runs on an encoding of the desired graph property. The symmetry propagator checks partially generated graphs for minimality with respect to a lexicographic ordering during the solving process. This approach has several advantages over a static symmetry breaking: (i) symmetries are detected early in the generation process, (ii) symmetry breaking is seamlessly integrated into the CDCL procedure, and (iii) the propagator performs a complete symmetry breaking without causing a prohibitively large initial encoding. We instantiate our approach by generating extremal graphs with certain restrictions in terms of forbidden subgraphs and diameter. In particular, we could confirm the Murty-Simon Conjecture (1979) on diameter-2-critical graphs for graphs up to 19 vertices and prove the exact number of Ramsey graphs \(\mathcal{R}(3,5,n)\) and \(\mathcal{R}(4,4,n)\) .

On vertex Ramsey graphs with forbidden subgraphs

A classical vertex Ramsey result due to Nešetřil and Rödl states that given a finite family of graphs F, a graph A and a positive integer r, if every graph B∈F has a 2-vertex-connected subgraph which is not a subgraph of A, then there exists an F-free graph which is vertex r-Ramsey with respect to A. We prove that this sufficient condition for the existence of an F-free graph which is vertex r-Ramsey with respect to A is also necessary for large enough number of colours r.We further show a generalisation of the result to a family of graphs and the typical existence of such a subgraph in a dense binomial random graph.

Shannon Entropy of Ramsey Graphs with up to Six Vertices.

Shannon entropy quantifying bi-colored Ramsey complete graphs is introduced and calculated for complete graphs containing up to six vertices. Complete graphs in which vertices are connected with two types of links, labeled as α-links and β-links, are considered. Shannon entropy is introduced according to the classical Shannon formula considering the fractions of monochromatic convex α-colored polygons with n α-sides or edges, and the fraction of monochromatic β-colored convex polygons with m β-sides in the given complete graph. The introduced Shannon entropy is insensitive to the exact shape of the polygons, but it is sensitive to the distribution of monochromatic polygons in a given complete graph. The introduced Shannon entropies Sα and Sβ are interpreted as follows: Sα is interpreted as an average uncertainty to find the green α-polygon in the given graph; Sβ is, in turn, an average uncertainty to find the red β-polygon in the same graph. The re-shaping of the Ramsey theorem in terms of the Shannon entropy is suggested. Generalization for multi-colored complete graphs is proposed. Various measures quantifying the Shannon entropy of the entire complete bi-colored graphs are suggested. Physical interpretations of the suggested Shannon entropies are discussed.

Maximum clique deleted from ramsey graphs of a graph and paths

Maximum clique deleted from ramsey graphs of a graph and paths

On Ramsey (C4, K1, n)-minimal graphs

Let F , G and H be any simple graphs. The notation F → ( G , H ) means for any red-blue coloring on the edges of graph F , there exists either a red copy of G or a blue copy of H . If F → ( G , H ) , then graph F is called a Ramsey graph for ( G , H ) . Additionally, if the graph F satisfies that F − e ↛ ( G , H ) for any edge e of F , then graph F is called a Ramsey ( G , H ) -minimal. The set of all Ramsey ( G , H ) -minimal graphs is denoted by ℛ( G , H ) . In this paper, we construct a new class of Ramsey ( C 4 , K 1, n ) -minimal graphs.

On the Minimum Degree of Minimal Ramsey Graphs for Cliques Versus Cycles

A graph is said to be -Ramsey for a -tuple of graphs , denoted by , if every -edge-coloring of contains a monochromatic copy of in color for some . Let denote the smallest minimum degree of over all graphs that are minimal -Ramsey for (with respect to subgraph inclusion). The study of this parameter was initiated in 1976 by Burr, Erdős, and Lovász, who determined its value precisely for a pair of cliques. Over the past two decades the parameter has been studied by several groups of authors, their main focus being on the symmetric case, where for all . The asymmetric case, in contrast, has received much less attention. In this paper, we make progress in this direction, studying asymmetric tuples consisting of cliques, cycles, and trees. We determine when is a pair of one clique and one tree, a pair of one clique and one cycle, and a pair of two different cycles. We also generalize our results to multiple colors and obtain bounds on in terms of the size of the cliques , the number of cycles, and the number of cliques. Our bounds are tight up to logarithmic factors when two of the three parameters are fixed.

Anticoncentration in Ramsey graphs and a proof of the Erdős–McKay conjecture

Abstract An n-vertex graph is called C-Ramsey if it has no clique or independent set of size $C\log _2 n$ (i.e., if it has near-optimal Ramsey behavior). In this paper, we study edge statistics in Ramsey graphs, in particular obtaining very precise control of the distribution of the number of edges in a random vertex subset of a C-Ramsey graph. This brings together two ongoing lines of research: the study of ‘random-like’ properties of Ramsey graphs and the study of small-ball probability for low-degree polynomials of independent random variables. The proof proceeds via an ‘additive structure’ dichotomy on the degree sequence and involves a wide range of different tools from Fourier analysis, random matrix theory, the theory of Boolean functions, probabilistic combinatorics and low-rank approximation. In particular, a key ingredient is a new sharpened version of the quadratic Carbery–Wright theorem on small-ball probability for polynomials of Gaussians, which we believe is of independent interest. One of the consequences of our result is the resolution of an old conjecture of Erdős and McKay, for which Erdős reiterated in several of his open problem collections and for which he offered one of his notorious monetary prizes.

Multicolor bipartite Ramsey numbers for quadrilaterals and stars

For bipartite graphs H1,…,Hμ, μ≥2, the μ-color bipartite Ramsey number, denoted by Rb(H1,…,Hμ), is the least positive integer N such that if we arbitrarily color the edges of a complete bipartite graph KN,N with μ colors, then it contains a monochromatic copy of Hi in color i for some i, 1≤i≤μ. Let C4 and K1,n be a quadrilateral and a star on n+1 vertices, respectively. In this paper, we show that the (μ+1)-color bipartite Ramsey number Rb(C4,…,C4,K1,n)≤n+⌈12μ2(4n+μ2+2μ−7)+4⌉+μ2+μ2−1. Moreover, using algebraic methods, we construct Ramsey graphs or near Ramsey graphs and determine infinitely many values of Rb(C4,…,C4,K1,n), which reach the upper bound if μ=1,2 and are at most ⌊μ2⌋ less than the upper bound if μ≥3.

Minimal Ramsey Graphs with Many Vertices of Small Degree

Given any graph $H$, a graph $G$ is said to be $q$-Ramsey for $H$ if every coloring of the edges of $G$ with $q$ colors yields a monochromatic subgraph isomorphic to $H$. Such a graph $G$ is said to be minimal $q$-Ramsey for $H$ if additionally no proper subgraph $G'$ of $G$ is $q$-Ramsey for $H$. In 1976, Burr, Erdös, and Lovász initiated the study of the parameter $s_q(H)$, defined as the smallest minimum degree among all minimal $q$-Ramsey graphs for $H$. In this paper, we consider the problem of determining how many vertices of degree $s_q(H)$ a minimal $q$-Ramsey graph for $H$ can contain. Specifically, we seek to identify graphs for which a minimal $q$-Ramsey graph can contain arbitrarily many such vertices. We call a graph satisfying this property $s_q$-abundant. Among other results, we prove that every cycle is $s_q$-abundant for any integer $q\geq 2$. We also discuss the cases when $H$ is a clique or a clique with a pendant edge, extending previous results of Burr and co-authors and Fox and co-authors. To prove our results and construct suitable minimal Ramsey graphs, we use gadget graphs, which we call pattern gadgets and which generalize earlier constructions used in the study of minimal Ramsey graphs. We provide a new, more constructive proof of the existence of these gadgets.

The minimum degree of minimal Ramsey graphs for cliques

We prove that $s_r(K_k) = O(k^5 r^{5/2})$, where $s_r(K_k)$ is the Ramsey parameter introduced by Burr, Erd\H{o}s and Lov\'{a}sz in 1976, which is defined as the smallest minimum degree of a graph $G$ such that any $r$-colouring of the edges of $G$ contains a monochromatic $K_k$, whereas no proper subgraph of $G$ has this property. The construction used in our proof relies on a group theoretic model of generalised quadrangles introduced by Kantor in 1980.

Special Section on the Forty-Ninth Annual ACM Symposium on the Theory of Computing (STOC 2017)

This issue of SICOMP contains ten specially selected papers from STOC 2017, the Forty-ninth Annual ACM Symposium on the Theory of Computing, which was held June 19--23 in Montreal, Canada. The papers here were chosen to represent the range and quality of the STOC program. These papers have been revised and extended by their authors and subjected to the standard thorough reviewing process of SICOMP. The program committee for STOC 2017 consisted of Nina Balcan, Allan Borodin, Keren Censor-Hillel, Edith Cohen, Artur Czumaj, Yevgeniy Dodis, Andrew Drucker, Nick Harvey, Monika Henzinger, Russell Impagliazzo, Ken-ichi Kawarabayashi, Ravi Kumar, James R. Lee, Katrina Ligett, Aleksander Mądry, Cristopher Moore, Jelani Nelson, Eric Price, Amit Sahai, Jared Saia, Shubhangi Saraf, Alexander Sherstov, Mohit Singh, and Gábor Tardos. The program chair was Valerie King. Included in this issue are the following papers: ``Short Presburger Arithmetic Is Hard," by Danny Nguyen and Igor Pak, proves that the satisfiability of short sentences in Presburger arithmetic with $m+2$ alternating quantifiers is $\Sigma^{{P}}_m$-complete or $\Pi^{{P}}_m$-complete when the first quantifier is $\exists$ or $\forall$, respectively. ``An Efficient Reduction from Two-Source to Nonmalleable Extractors: Achieving Near-Logarithmic Min-Entropy," by Avraham Ben-Aroya, Dean Doron, and Amnon Ta-Shma, gets an explicit bipartite Ramsey graph (or a twosource extractor) for sets of size 2$k$ for $k = O(\log n \log \log n)$, using the currently best explicit nonmalleable extractors. ``Holographic Algorithm with Matchgates Is Universal for Planar \#CSP over Boolean Domain," by Jin-Yi Cai and Zhiguo Fu, classifies all counting CSPs over Boolean variables into one of three categories: polynomial-time tractable, \#P-hard for general instances but solvable in polynomial time over planar graphs, and \#P-hard over planar graphs. ``Deciding Parity Games in Quasipolynomial Time," by Cristian S. Calude, Sanjay Jain, Bakhadyr Khoussainov, Wei Li, and Frank Stephan, shows the parameterized parity game, with $n$ nodes and $m$ priorities, is in the class of fixed parameter tractable problems when parameterized over $m$. ``New Hardness Results for Routing on Disjoint Paths," by Julia Chuzhoy, David H. K. Kim, and Rachit Nimavat, proves that node-disjoint paths is $2^{\Omega(\sqrt{\log n})}$-hard to approximate, unless all problems in NP have algorithms with running time $n^{O(\log n)}$. ``A Weighted Linear Matroid Parity Algorithm," by Satoru Iwata and Yusuke Kobayashi, presents a combinatorial, deterministic, strongly polynomial-time algorithm for the weighted linear matroid parity problem. ``Targeted Pseudorandom Generators, Simulation Advice Generators, and Derandomizing Logspace," by William M. Hoza and Chris Umans, shows that $\mathbf{BPL} \subseteq \bigcap_{\alpha > 0} {DSPACE}(\log^{1 + \alpha} n)$, assuming that for every derandomization result for log-space algorithms there is a pseudorandom generator strong enough to nearly recover the derandomization by iterating over all seeds and taking a majority vote. ``Approximating Rectangles by Juntas and Weakly Exponential Lower Bounds for LP Relaxations of CSPs," by Pravesh K. Kothari, Raghu Meka, and Prasad Raghavendra, shows that for CSPs, subexponential size LP relaxations are as powerful as $n^{\Omega(1)}$-rounds of the Sherali--Adams LP hierarchy. ``Equivocating Yao: Constant-Round Adaptively Secure Multiparty Computation in the Plain Model," by Ran Canetti, Oxana Poburinnaya, and Muthuramakrishnan Venkitasubramaniam, defines a new type of encryption and shows that Yao's garbling scheme, implemented with this encryption mechanism, is secure against adaptive adversaries. ``Geodesic Walks in Polytopes," by Yin Tat Lee and Santosh Vempala, introduces the geodesic walk for sampling Riemannian manifolds and applies it to the problem of generating uniform random points from the interior of polytopes in ${\mathbb{R}}^{n}$ specified by m inequalities; the resulting sampling algorithm for polytopes mixes in $O^{*}(mn^{\frac{3}{4}})$ steps. We thank the authors, the STOC 2017 program committee, the STOC 2017 external reviewers, and the SICOMP referees for all of their hard work. Andy Drucker, Ravi Kumar, Amit Sahai, Mohit Singh, Guest editors

New bounds for Ramsey numbers [formula omitted

Let R(H1,H2) denote the Ramsey number for the graphs H1,H2, and let Jk be Kk−e. We present algorithms which enumerate all circulant and block-circulant Ramsey graphs for different types of graphs, thereby obtaining several new lower bounds on Ramsey numbers including: 49≤R(K3,J12), 36≤R(J4,K8), 43≤R(J4,J10), 52≤R(K4,J8), 37≤R(J5,J6), 43≤R(J5,K6), 65≤R(J5,J7). We also use a gluing strategy to derive a new upper bound on R(J5,J6). With both strategies combined, we prove the value of two Ramsey numbers: R(J5,J6)=37 and R(J5,J7)=65. We also show that the 64-vertex extremal Ramsey graph for R(J5,J7) is unique. Furthermore, our algorithms also allow the establishment of new lower bounds and exact values on Ramsey numbers involving wheel graphs and complete bipartite graphs, including: R(W7,W4)=21, R(W7,W7)=19, R(K3,4,K3,4)=25, and R(K3,5,K3,5)=33.

Modeling of complex ecosystems, extension of deserts and violation of the short water cycle

The large ecosystems (region, country, and planet) have many properties common with complex systems. The main such characteristics include dynamic instability (e.g. weather dependence), weak causality (many possible causes per event) and event driven behavior (especially in arid and semi-arid areas). This paper focuses attention on modeling ecosystems by approaches used for complex systems.When modeling complex ecosystems, we usually encounter the following difficulties: partiality, large amounts of data and uncertainties of conclusions. It can be said that none of the known approaches solves these difficulties perfectly. The most common is the physical approach, sometimes reinforced by statistical procedures. The physical approach to modeling leads to a complicated description of phenomena associated with relatively simple geometry. A complicated description usually requires a large amount of data (measured or simulated) and thus more complicated calculations. If we assume emergences in the ecosystem, a physical approach is not appropriate at all. In the presented article we apply the approach of so-called structural invariants, which has the opposite properties: a simple description of phenomena associated with a more complex geometry (in our case pre geometry). It does not require as much data and calculations are simple. The price paid is a qualitative interpretation of the results, which carries a special type of uncertainty. The structural invariant used in the article is the invariant Matroid and Bases of Matroid (M, BM) in combination with Ramsey graph theory. In addition, the article introduces a calculus that describes the emergent phenomenon using two quantities - the power of the emergent phenomenon and the complexity of the structure that is associated with this phenomenon. The developed method is used for a novel application of modeling the process of desertification of Earth. In this approach, we understand desertification as an emergent evolutionary operation of the Earth's development. In the sequence of two large previous emergences (warming of the Earth about 11,700 years ago and drying of the earth beginning about6000 years ago), the time of possible further emergences related to the “desert expansion” operation is calculated. The second application of the method is the analysis of the operation “violation of Short Water Cycle” in the landscape and its possible contribution to the desertification.

Modeling Complex Systems by Structural Invariants Approach

When modeling complex systems, we usually encounter the following difficulties: partiality, large amount of data, and uncertainty of conclusions. It can be said that none of the known approaches solves these difficulties perfectly, especially in cases where we expect emergences in the complex system. The most common is the physical approach, sometimes reinforced by statistical procedures. The physical approach to modeling leads to a complicated description of phenomena associated with a relatively simple geometry. If we assume emergences in the complex system, the physical approach is not appropriate at all. In this article, we apply the approach of structural invariants, which has the opposite properties: a simple description of phenomena associated with a more complicated geometry (in our case pregeometry). It does not require as much data and the calculations are simple. The price paid for the apparent simplicity is a qualitative interpretation of the results, which carries a special type of uncertainty. Attention is mainly focused (in this article) on the invariant matroid and bases of matroid (M, BM) in combination with the Ramsey graph theory. In addition, this article introduces a calculus that describes the emergent phenomenon using two quantities—the power of the emergent phenomenon and the complexity of the structure that is associated with this phenomenon. The developed method is used in the paper for modeling and detecting emergent situations in cases of water floods, traffic jams, and phase transition in chemistry.

Large cliques and independent sets all over the place

We study the following question raised by Erdős and Hajnal in the early 90’s. Over all n n -vertex graphs G G what is the smallest possible value of m m for which any m m vertices of G G contain both a clique and an independent set of size log ⁡ n \log n ? We construct examples showing that m m is at most 2 2 ( log ⁡ log ⁡ n ) 1 / 2 + o ( 1 ) 2^{2^{(\log \log n)^{1/2+o(1)}}} obtaining a twofold sub-polynomial improvement over the upper bound of about n \sqrt {n} coming from the natural guess, the random graph. Our (probabilistic) construction gives rise to new examples of Ramsey graphs, which while having no very large homogenous subsets contain both cliques and independent sets of size log ⁡ n \log n in any small subset of vertices. This is very far from being true in random graphs. Our proofs are based on an interplay between taking lexicographic products and using randomness.

Special Section on the 48th Annual ACM Symposium on Theory of Computing (STOC 2016)

Special Section on the 48th Annual ACM Symposium on Theory of Computing (STOC 2016)

Phase Transition as an Emergent Phenomenon Analysed by Violation of Structural Invariant (M, BM)

When modeling complex systems, we usually encounter the following difficulties: partiality, large amounts of data and uncertainty of conclusions. The most common approach used for modeling is the physical approach, sometimes reinforced by statistical procedures. If we assume emergences in the complex system, a physical approach is not appropriate at all. Instead, we build here the approach of structural invariants. In this paper, we show that another plane can be built above the plane of physical description, which is responsible for violation of structural invariants. Main attention is concentrated (in this article) on the invariant matroid and bases of matroid (M, BM) in combination with Ramsey graph theory. In addition, the article introduces a calculus that describes the emergent phenomena using two quantities - the power of the emergent phenomenon and the complexity of the structure of the considered complex system. We show the application of the method for modeling phase transition in chemistry.

Complete bipartite graphs deleted in Ramsey graphs

For graphs F, G and H, let F→(G,H) signify that any red/blue edge coloring of F contains either a red G or a blue H. The Ramsey number R(G,H) is defined as min⁡{r|Kr→(G,H)}. In this note, we consider an optimization problem to bound the complete bipartite-critical Ramsey number RΛ(G,H) defined as max⁡{t|Kr∖Kt,t→(G,H)} where r=R(G,H) and Λ is a set of Kt,t, and determine RΛ(G,H) for some pairs (G,H).