scholarly journals A Ramsey Theorem for Biased Graphs

2020 ◽  
Vol 34 (4) ◽  
pp. 2270-2281
Author(s):  
Peter Nelson ◽  
Sophia Park
Keyword(s):  
Ramsey Theorem ◽  
Biased Graphs

scholarly journals A Canonical Ramsey Theorem for Exactly m-Coloured Complete Subgraphs

2013 ◽  
Vol 23 (1) ◽  
pp. 102-115 ◽  
Author(s):  
TEERADEJ KITTIPASSORN ◽  
BHARGAV P. NARAYANAN
Keyword(s):  
Natural Number ◽  
Complete Graph ◽  
Infinite Subset ◽  
Complete Subgraph ◽  
Ramsey Theorem ◽  
Distinguished Vertex ◽  
Edge Colourings ◽  
Distinct Colour

Given an edge colouring of a graph with a set of m colours, we say that the graph is exactly m-coloured if each of the colours is used. We consider edge colourings of the complete graph on $\mathbb{N}$ with infinitely many colours and show that either one can find an exactly m-coloured complete subgraph for every natural number m or there exists an infinite subset X ⊂ $\mathbb{N}$ coloured in one of two canonical ways: either the colouring is injective on X or there exists a distinguished vertex v in X such that X\{v} is 1-coloured and each edge between v and X\{v} has a distinct colour (all different to the colour used on X\{v}). This answers a question posed by Stacey and Weidl in 1999. The techniques that we develop also enable us to resolve some further questions about finding exactly m-coloured complete subgraphs in colourings with finitely many colours.

A Quasi Ramsey Theorem

2013 ◽  
pp. 111-118
Author(s):  
Hans Jürgen Prömel
Keyword(s):  
Ramsey Theorem

scholarly journals Detection of Emergent Situations in Complex Systems by Structural Invariant (MB, M)

MENDEL ◽  
2017 ◽  
Vol 23 (1) ◽  
pp. 163-170 ◽  
Author(s):  
Jiri Bila ◽  
Martin Novak
Keyword(s):  
Complex System ◽  
Complex Systems ◽  
Detection Method ◽  
Water Cycle ◽  
Ramsey Theorem ◽  
Structural Invariant

The paper introduces complete description of the detection method that uses structural invariant Matroid and its Bases (MB, M). There are recapitulated essential concepts from the used knowledge field as “complex system, emergent situations (A, B, C)”, Ramsey theorem and principal computation variables “power” and “complexity” of emergence phenomenon. The method is explained in details and the demonstration of its application is done by the detection of emergent situation – violation of Short Water Cycle in an ecosystem.

scholarly journals Biased graphs. VI. synthetic geometry

2019 ◽  
Vol 81 ◽  
pp. 119-141
Author(s):  
Rigoberto Flórez ◽  
Thomas Zaslavsky
Keyword(s):  
Biased Graphs

scholarly journals Constrained Ramsey Numbers

2009 ◽  
Vol 18 (1-2) ◽  
pp. 247-258 ◽  
Author(s):  
PO-SHEN LOH ◽  
BENNY SUDAKOV
Keyword(s):  
Complete Graph ◽  
Ramsey Number ◽  
Ramsey Numbers ◽  
Ramsey Theorem ◽  
Logarithmic Factor ◽  
Subgraph Isomorphic ◽  
Rate Of Growth ◽  
Special Cases ◽  
Edge Colouring

For two graphs S and T, the constrained Ramsey number f(S, T) is the minimum n such that every edge colouring of the complete graph on n vertices (with any number of colours) has a monochromatic subgraph isomorphic to S or a rainbow subgraph isomorphic to T. Here, a subgraph is said to be rainbow if all of its edges have different colours. It is an immediate consequence of the Erdős–Rado Canonical Ramsey Theorem that f(S, T) exists if and only if S is a star or T is acyclic. Much work has been done to determine the rate of growth of f(S, T) for various types of parameters. When S and T are both trees having s and t edges respectively, Jamison, Jiang and Ling showed that f(S, T) ≤ O(st2) and conjectured that it is always at most O(st). They also mentioned that one of the most interesting open special cases is when T is a path. In this paper, we study this case and show that f(S, Pt) = O(st log t), which differs only by a logarithmic factor from the conjecture. This substantially improves the previous bounds for most values of s and t.

SPARSE PARTITION REGULARITY

2006 ◽  
Vol 93 (3) ◽  
pp. 545-569 ◽  
Author(s):  
IMRE LEADER ◽  
PAUL A. RUSSELL
Keyword(s):  
Chromatic Number ◽  
Independent Interest ◽  
Natural Numbers ◽  
Ramsey Theorem

Our aim in this paper is to prove Deuber's conjecture on sparse partition regularity, that for every $m$, $p$ and $c$ there exists a subset of the natural numbers whose $(m,p,c)$-sets have high girth and chromatic number. More precisely, we show that for any $m$, $p$, $c$, $k$ and $g$ there is a subset $S$ of the natural numbers that is sufficiently rich in $(m,p,c)$-sets that whenever $S$ is $k$-coloured there is a monochromatic $(m,p,c)$-set, yet is so sparse that its $(m,p,c)$-sets do not form any cycles of length less than $g$.Our main tools are some extensions of Nešetřil–Rödl amalgamation and a Ramsey theorem of Bergelson, Hindman and Leader. As a sideline, we obtain a Ramsey theorem for products of trees that may be of independent interest.

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