Ramsey's theorem ≈ Ramsey's theorem
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On the Multi-Colored Ramsey Numbers of Paths and Even Cycles Open
In this paper we improve the upper bound on the multi-color Ramsey numbers of paths and even cycles. More precisely, we prove the following. For every $r\geq 2$ there exists an $n_0=n_0(r)$ such that for $n\geq n_0$ we have $$R_r(P_n) \leq…
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Improved Bounds on the Multicolor Ramsey Numbers of Paths and Even Cycles Open
We study the multicolor Ramsey numbers for paths and even cycles, $R_k(P_n)$ and $R_k(C_n)$, which are the smallest integers $N$ such that every coloring of the complete graph $K_N$ has a monochromatic copy of $P_n$ or $C_n$ respectively. …
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Lower bounds for Ramsey numbers as a statistical physics problem Open
Ramsey’s theorem, concerning the guarantee of certain monochromatic patterns in large enough edge-coloured complete graphs, is a fundamental result in combinatorial mathematics. In this work, we highlight the connection between this abstra…
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The size Ramsey number of short subdivisions of bounded degree graphs Open
For graphs G and F , write if any coloring of the edges of G with colors yields a monochromatic copy of the graph F . Suppose is obtained from a graph S with s vertices and maximum degree d by subdividing its edges h times (that is, by rep…
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A Generalization of Schur's Theorem Open
This paper is an excerpt from the author's 1968 PhD dissertation [Yale University, 1968] in which the (now) well-known result, commonly known as the Folkman-Rado-Sanders theorem, is proved. The proof uses (finite) alternating sums of integ…
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New Lower Bounds on the Size-Ramsey Number of a Path Open
We prove that for all graphs with at most $(3.75-o(1))n$ edges there exists a 2-coloring of the edges such that every monochromatic path has order less than $n$. This was previously known to be true for graphs with at most $2.5n-7.5$ edges…
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An exponential improvement for diagonal Ramsey Open
The Ramsey number $R(k)$ is the minimum $n \in \mathbb{N}$ such that every red-blue colouring of the edges of the complete graph $K_n$ on $n$ vertices contains a monochromatic copy of $K_k$. We prove that \[ R(k) \leqslant (4 - \varepsilon…
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Size multipartite Ramsey numbers for stripes versus small cycles Open
For simple graphs $G_1$ and $G_2$, the size Ramsey multipartite number $m_j(G_1, G_2)$ is defined as the smallest natural number $s$ such that any arbitrary two coloring of the graph $K_{j \times s}$ using the colors red and blue, contains…
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The Ramsey theory of Henson graphs Open
Analogues of Ramsey’s Theorem for infinite structures such as the rationals or the Rado graph have been known for some time. In this context, one looks for optimal bounds, called degrees, for the number of colors in an isomorphic substruct…
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On size multipartite Ramsey numbers for stars versus paths and cycles Open
Let $K_{l\\times t}$ be a complete, balanced, multipartite graph consisting of $l$ partite sets and $t$ vertices in each partite set. For given two graphs $G_1$ and $G_2$, and integer $j\\geq 2$, the size multipartite Ramsey number $m_j(G_…
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One More Turán Number and Ramsey Number for the Loose 3-Uniform Path of Length Three Open
Let P denote a 3-uniform hypergraph consisting of 7 vertices a, b, c, d, e, f, g and 3 edges {a, b, c}, {c, d, e}, and {e, f, g}. It is known that the r-color Ramsey number for P is R(P; r) = r + 6 for r ≤ 9. The proof of this result relie…
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THE DEFINABILITY STRENGTH OF COMBINATORIAL PRINCIPLES Open
We introduce the definability strength of combinatorial principles. In terms of definability strength, a combinatorial principle is strong if solving a corresponding combinatorial problem could help in simplifying the definition of a defin…
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Ramsey properties of randomly perturbed dense graphs Open
We investigate Ramsey properties of a random graph model in which random edges are added to a given dense graph. Specifically, we determine lower and upper bounds on the function $p=p(n)$ that ensures that for any dense graph $G_n$ a.a.s. …
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On Ramsey (4K2,P3)-minimal graphs Open
Let F , G , and H be simple graphs. We write F → (G , H) to mean that any red–blue coloring of all edges of F will contain either a red copy of G or a blue copy of H . A graph F (without isolated vertices) satisfying F → (G , H) and for ea…
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Ramsey Numbers of Berge-Hypergraphs and Related Structures Open
For a graph $G=(V,E)$, a hypergraph $\mathcal{H}$ is called a Berge-$G$, denoted by $BG$, if there exists an injection $f: E(G) \to E(\mathcal{H})$ such that for every $e \in E(G)$, $e \subseteq f(e)$. Let the Ramsey number $R^r(BG,BG)$ be…
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Restricted size Ramsey number for path of order three versus graph of order five Open
Let $G$ and $H$ be simple graphs. The Ramsey number for a pair of graph $G$ and $H$ is the smallest number $r$ such that any red-blue coloring of edges of $K_r$ contains a red subgraph $G$ or a blue subgraph $H$. The size Ramsey number for…
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A note on Ramsey numbers for Berge-G hyper graphs Open
For a graph G=(V,E), a hypergraph H is called Berge-G if there is a bijection f from E(G) to E(H) such that for each e in E(G), e is a subset of f(e). The set of all Berge-G hypergraphs is denoted B(G). For integers k>1, r>1, and a graph G…
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Conant's generalised metric spaces are Ramsey Open
We give Ramsey expansions of classes of generalised metric spaces where distances come from a linearly ordered commutative monoid. This complements results of Conant about the extension property for partial automorphisms and extends an ear…
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A General Lower Bound on Gallai-Ramsey Numbers for Non-Bipartite Graphs Open
Given a graph $H$ and a positive integer $k$, the $k$-color Gallai-Ramsey number $gr_{k}(K_{3} : H)$ is defined to be the minimum number of vertices $n$ for which any $k$-coloring of the complete graph $K_{n}$ contains either a rainbow tri…
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New bounds on the strength of some restrictions of Hindman’s Theorem Open
The relations between (restrictions of) Hindman’s Finite Sums Theorem and (variants of) Ramsey’s Theorem give rise to long-standing open problems in combinatorics, computability theory and proof theory. We present some results motivated by…
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The proof-theoretic strength of Ramsey's theorem for pairs and two colors Open
Ramsey's theorem for $n$-tuples and $k$-colors ($\mathsf{RT}^n_k$) asserts that every k-coloring of $[\mathbb{N}]^n$ admits an infinite monochromatic subset. We study the proof-theoretic strength of Ramsey's theorem for pairs and two color…
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Ramsey Algebras and Formal Orderly Terms Open
Hindman’s theorem says that every finite coloring of the natural numbers has a monochromatic set of finite sums. A Ramsey algebra is a structure that satisfies an analogue of Hindman’s theorem. In this paper, we present the basic notions o…
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Multicolor Ramsey Numbers via Pseudorandom Graphs Open
A weakly optimal $K_s$-free $(n,d,\lambda)$-graph is a $d$-regular $K_s$-free graph on $n$ vertices with $d=\Theta(n^{1-\alpha})$ and spectral expansion $\lambda=\Theta(n^{1-(s-1)\alpha})$, for some fixed $\alpha>0$. Such a graph is called…
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The reverse mathematics of Ramsey-type theorems Open
In this thesis, we investigate the computational content and the logical strength of Ramsey's theorem and its consequences. For this, we use the frameworks of reverse mathematics and of computable reducibility. We proceed to a systematic s…
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Maximum number of triangle-free edge colourings with five and six colours Open
Let k ≥ 3 and r ≥ 2 be natural numbers. For a graph G, let F(G, k, r) denote the number of colourings of the edges of G with colours 1,…, r such that, for every colour c ∈ {1,…, r}, the edges of colour c contain no complete graph on k vert…
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Actions on semigroups and an infinitary Gowers–Hales–Jewett Ramsey theorem Open
We introduce the notion of (Ramsey) action on a (filtered) semigroup. We then prove in this setting a general result providing a common generalization of the infinitary Gowers Ramsey theorem for multiple tetris operations, the infinitary H…
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On the Maximum Number of Integer Colourings with Forbidden Monochromatic Sums Open
Let $f(n,r)$ denote the maximum number of colourings of $A \subseteq \lbrace 1,\ldots,n\rbrace$ with $r$ colours such that each colour class is sum-free. Here, a sum is a subset $\lbrace x,y,z\rbrace$ such that $x+y=z$. We show that $f(n,2…
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Improved bounds on the multicolor Ramsey numbers of paths and even cycles Open
We study the multicolor Ramsey numbers for paths and even cycles, Rk(Pn) and Rk(Cn), which are the smallest integers N such that every coloring of the complete graph KN has a monochromatic copy of Pn or Cn respectively. For a long time, Rk…
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The edge Folkman number $F_e(3, 3; 4)$ is greater than 19 Open
The set of the graphs which do not contain the complete graph on $q$ vertices $K_q$ and have the property that in every coloring of their edges in two colors there exist a monochromatic triangle is denoted by $\mathcal{H}_e(3, 3; q)$. The …
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Multicolor star-critical Ramsey numbers and Ramsey-good graphs Open
This paper seeks to develop the multicolor version of star-critical Ramsey numbers, which serve as a measure of the strength of the corresponding Ramsey numbers. We offer several general theorems, some of which focus on Ramsey-good cases (…