A Euclidean Ramsey Problem

   Abstract. Let C _n denote the set of points in R ⁿ whose coordinates are all 0 or 1 , i.e., the vertex set of the unit n -cube. Graham and Rothschild [2] proved that there exists an integer N such that for n ≥ N , any 2-coloring of the edges of the complete graph on C _n contains a monochromatic plane K ₄ . Let N ^* be the minimum such N . They noted that N ^* must be at least 6 . Their upper bound on N ^* has come to be known as Graham's number , often cited as the largest number that has ever been put to any practical use. In this note we show that N ^* must be at least 11 and provide some experimental evidence suggesting that N ^* is larger still.     

A Euclidean Ramsey Problem

Abstract. Let C _n denote the set of points in R ⁿ whose coordinates are all 0 or 1 , i.e., the vertex set of the unit n -cube. Graham and Rothschild [2] proved that there exists an integer N such that for n ≥ N , any 2-coloring of the edges of the complete graph on C _n contains a monochromatic plane K ₄ . Let N ^* be the minimum such N . They noted that N ^* must be at least 6 . Their upper bound on N ^* has come to be known as Graham's number , often cited as the largest number that has ever been put to any practical use. In this note we show that N ^* must be at least 11 and provide some experimental evidence suggesting that N ^* is larger still.     

A result in asymmetric Euclidean Ramsey theory

It is proved that if the points of the three-dimensional Euclidean space are coloured with red and blue, then there exist either two red points at unit distance, or six collinear blue points with distance one between any two consecutive points.     

Transitive sets in Euclidean Ramsey theory

A finite set X in some Euclidean space Rn is called Ramsey if for any k there is a d such that whenever Rd is k-coloured it contains a monochromatic set congruent to X. This notion was introduced by Erd?s, 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’ conjecture to the conjecture above. Calling a finite set transitive if its symmetry group acts transitively—in other words, if all points of the set look the same—our conjecture is that the Ramsey sets are precisely the transitive sets, together with their subsets. One appealing feature of this conjecture is that it reduces (in one direction) to a purely combinatorial statement. We give this statement as well as several other related conjectures. We also prove the first non-trivial cases of the statement.Curiously, it is far from obvious that our new conjecture is genuinely different from the old. We show that they are indeed different by proving that not every spherical set embeds in a transitive set. This result may be of independent interest.     

欧氏空间中紧致超曲面的欧氏直径的一点注记

本文对欧氏空间中分别具有给定截面曲率和Ｒｉｃｃｉ曲率的紧致超曲面的欧氏直径做了估计     

Hypergraph list coloring and Euclidean Ramsey theory

A hypergraph is simple if it has no two edges sharing more than a single vertex. It is s‐list colorable (or s‐choosable) if for any assignment of a list of s colors to each of its vertices, there is a vertex coloring assigning to each vertex a color from its list, so that no edge is monochromatic. We prove that for every positive integer r, there is a function d_r(s) such that no r‐uniform simple hypergraph with average degree at least d_r(s) is s‐list‐colorable. This extends a similar result for graphs, due to the first author, but does not give as good estimates of d_r(s) as are known for d₂(s), since our proof only shows that for each fixed r ≥ 2, d_r(s) ≤ 2 We use the result to prove that for any finite set of points X in the plane, and for any finite integer s, one can assign a list of s distinct colors to each point of the plane so that any coloring of the plane that colors each point by a color from its list contains a monochromatic isometric copy of X. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011     

A Note on Minimal Surfaces in Euclidean 3-Space

In this note, a construction of minimal surfaces in Euclidean 3-space is given. By using the product of Weierstrass data of two known minimal surfaces, one gets a new Weierstrass data and a corresponding minimal surface from the Weierstrass representation.     

A Result in Dual Ramsey Theory

We present a result which is obtained by combining a result of Carlson with the Finitary Dual Ramsey Theorem of Graham–Rothschild.     

A Note on Geometric Embeddings of Simplicial Complexes in a Euclidean Space

In this note we prove that if a simplicial complex K can be embedded geometrically in R ^m , then a certain linear system of equations associated with K possesses a small integral solution. Received July 5, 1998, and in revised form May13, 1999.     

Symmetrized Induced Ramsey Theory

We prove induced Ramsey theorems in which the monochromatic induced subgraph satisfies that all members of a prescribed set of its partial isomorphisms extend to automorphisms of the colored graph (without requirement of preservation of colors). We consider vertex and edge colorings, and extensions of partial isomorphisms in the set of all partial isomorphisms between singletons as considered by Babai and Sós (European J Combin 6(2):101–114, 1985), the set of all finite partial isomorphisms as considered by Hrushovski (Combinatorica 12(4):411–416, 1992), Herwig (Combinatorica 15:365–371, 1995) and Herwig-Lascar (Trans Amer Math Soc 5:1985–2021, 2000), and the set of all total isomorphisms. We observe that every finite graph embeds into a finite vertex transitive graph by a so called bi-embedding, an embedding that is compatible with a monomorphism between the corresponding automorphism groups. We also show that every countable graph bi-embeds into Rado’s universal countable graph Γ.     

Rainbow Generalizations of Ramsey Theory: A Survey

In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs.     

A Note on Multiplicative Ideal Theory

We state a semigroup version of (19.15)THEOREM and (36.2)THEOREM, and disprove (30.9)THEOREM of Gilmers book [3] for commutative semigroups. We give a condition for the Kronecker function ring D* with respect to a star-operation * on an integral domain D to be well-defined.2000 Mathematics Subject Classification: Primary 13A15 Secondary 20M14.     

Choiceless Ramsey Theory of Linear Orders

Motivated by work of Erd?s, Milner and Rado, we investigate symmetric and asymmetric partition relations for linear orders without the axiom of choice. The relations state the existence of a subset in one of finitely many given order types that is homogeneous for a given colouring of the finite subsets of a fixed size of a linear order. We mainly study the linear orders 〈 ^α 2,< _{l e x} 〉, where α is an infinite ordinal and < _{l e x} is the lexicographical order. We first obtain the consistency of several partition relations that are incompatible with the axiom of choice. For instance we derive partition relations for 〈 ^ω 2,< _{l e x} 〉 from the property of Baire for all subsets of ^ω 2 and show that the relation \(\langle ^{\kappa }{2}, <_{lex}\rangle \longrightarrow (\langle ^{\kappa }{2}, <_{lex}\rangle )^{2}_{2}\) is consistent for uncountable regular cardinals κ with κ ^<κ = κ. We then prove a series of negative partition relations with finite exponents for the linear orders 〈 ^α 2,< _{l e x} 〉. We combine the positive and negative results to completely classify which of the partition relations \(\langle ^{\omega }{2}, <_{lex}\rangle \longrightarrow (\bigvee _{\nu <\lambda }K_{\nu },\bigvee _{\nu <\mu }M_{\nu })^{m}\) for linear orders K _ν ,M _ν and m≤4 and 〈 ^ω 2,< _{l e x} 〉→(K,M) ⁿ for linear orders K,M and natural numbers n are consistent.