scholarly journals Finding Unavoidable Colorful Patterns in Multicolored Graphs

10.37236/8184 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Matt Bowen ◽  
Ander Lamaison ◽  
Alp Müyesser
Keyword(s):  
Natural Number ◽  
Complete Graph ◽  
Type Problem ◽  
Polish Spaces ◽  
Subgraph Isomorphic ◽  
Simple Characterization ◽  
Color Class ◽  
Ramsey Type

We provide multicolored and infinite generalizations for a Ramsey-type problem raised by Bollobás, concerning colorings of $K_n$ where each color is well-represented. Let $\chi$ be a coloring of the edges of a complete graph on $n$ vertices into $r$ colors. We call $\chi$ $\varepsilon$-balanced if all color classes have $\varepsilon$ fraction of the edges. Fix some graph $H$, together with an $r$-coloring of its edges. Consider the smallest natural number $R_\varepsilon^r(H)$ such that for all $n\geq R_\varepsilon^r(H)$, all $\varepsilon$-balanced colorings $\chi$ of $K_n$ contain a subgraph isomorphic to $H$ in its coloring. Bollobás conjectured a simple characterization of $H$ for which $R_\varepsilon^2(H)$ is finite, which was later proved by Cutler and Montágh. Here, we obtain a characterization for arbitrary values of $r$, as well as asymptotically tight bounds. We also discuss generalizations to graphs defined on perfect Polish spaces, where the corresponding notion of balancedness is each color class being non-meagre. 

scholarly journals Partitioning the Power Set of $[n]$ into $C_k$-free parts

10.37236/8385 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Eben Blaisdell ◽  
András Gyárfás ◽  
Robert A. Krueger ◽  
Ronen Wdowinski
Keyword(s):  
Upper Bound ◽  
Connected Graph ◽  
Type Problem ◽  
Color Class ◽  
Power Set ◽  
Ramsey Type ◽  
Partition Class ◽  
Cycle Path

We show that for $n \geq 3, n\ne 5$, in any partition of $\mathcal{P}(n)$, the set of all subsets of $[n]=\{1,2,\dots,n\}$, into $2^{n-2}-1$ parts, some part must contain a triangle — three different subsets $A,B,C\subseteq [n]$ such that $A\cap B,A\cap C,B\cap C$ have distinct representatives. This is sharp, since by placing two complementary pairs of sets into each partition class, we have a partition into $2^{n-2}$ triangle-free parts.  We also address a more general Ramsey-type problem: for a given graph $G$, find (estimate) $f(n,G)$, the smallest number of colors needed for a coloring of $\mathcal{P}(n)$, such that no color class contains a Berge-$G$ subhypergraph. We give an upper bound for $f(n,G)$ for any connected graph $G$ which is asymptotically sharp when $G$ is a cycle, path, or star. Additional bounds are given when $G$ is a $4$-cycle and when $G$ is a claw.

scholarly journals Band description of knots and Vassiliev invariants

2002 ◽  
Vol 133 (2) ◽  
pp. 325-343 ◽  
Author(s):  
KOUKI TANIYAMA ◽  
AKIRA YASUHARA
Keyword(s):  
Natural Number ◽  
Algebraic Condition ◽  
Vassiliev Invariants

In the 1990s, Habiro defined Ck-move of oriented links for each natural number k [5]. A Ck-move is a kind of local move of oriented links, and two oriented knots have the same Vassiliev invariants of order [les ] k−1 if and only if they are transformed into each other by Ck-moves. Thus he has succeeded in deducing a geometric conclusion from an algebraic condition. However, this theorem appears only in his recent paper [6], in which he develops his original clasper theory and obtains the theorem as a consequence of clasper theory. We note that the ‘if’ part of the theorem is also shown in [4], [9], [10] and [16], and in [13] Stanford gives another characterization of knots with the same Vassiliev invariants of order [les ] k−1.

scholarly journals Ramsey Numbers for Theta Graphs

2011 ◽  
Vol 2011 ◽  
pp. 1-9
Author(s):  
M. M. M. Jaradat ◽  
M. S. A. Bataineh ◽  
S. M. E. Radaideh
Keyword(s):  
Complete Graph ◽  
Ramsey Number ◽  
Ramsey Numbers ◽  
Subgraph Isomorphic

The graph Ramsey number is the smallest integer with the property that any complete graph of at least vertices whose edges are colored with two colors (say, red and blue) contains either a subgraph isomorphic to all of whose edges are red or a subgraph isomorphic to all of whose edges are blue. In this paper, we consider the Ramsey numbers for theta graphs. We determine , for . More specifically, we establish that for . Furthermore, we determine for . In fact, we establish that if is even, if is odd.

scholarly journals A Ramsey-type problem on right-angled triangles in space

1996 ◽  
Vol 150 (1-3) ◽  
pp. 61-67 ◽  
Author(s):  
Miklós Bóna ◽  
Géza Tóth
Keyword(s):  
Type Problem ◽  
Ramsey Type

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.

scholarly journals Spectral characterization of the complete graph removing a path of small length

2019 ◽  
Vol 257 ◽  
pp. 260-268 ◽  
Author(s):  
Lihuan Mao ◽  
Sebastian M. Cioabă ◽  
Wei Wang
Keyword(s):  
Complete Graph ◽  
Spectral Characterization ◽  
Small Length

scholarly journals Another characterization of congruence distributive varieties

2020 ◽  
Vol 57 (3) ◽  
pp. 284-289
Author(s):  
Paolo Lipparini
Keyword(s):  
Natural Number ◽  
Congruence Identity ◽  
Congruence Distributive

AbstractWe provide a Maltsev characterization of congruence distributive varieties by showing that a variety 𝓥 is congruence distributive if and only if the congruence identity … (k factors) holds in 𝓥, for some natural number k.

scholarly journals A strict upper bound for size multipartite Ramsey numbers of paths versus stars

2017 ◽  
Vol 1 (2) ◽  
pp. 9
Author(s):  
Chula Jayawardene
Keyword(s):  
Natural Number ◽  
Complete Graph ◽  
Upper Bound ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Infinite Class ◽  
Ramsey Numbers ◽  
Necessary And Sufficient

<p>Let $P_n$ represent the path of size $n$. Let $K_{1,m-1}$ represent a star of size $m$ and be denoted by $S_{m}$. Given a two coloring of the edges of a complete graph $K_{j \times s}$ we say that $K_{j \times s}\rightarrow (P_n,S_{m+1})$ if there is a copy of $P_n$ in the first color or a copy of $S_{m+1}$ in the second color. The size Ramsey multipartite number $m_j(P_n, S_{m+1})$ is the smallest natural number $s$ such that $K_{j \times s}\rightarrow (P_n,S_{m+1})$. Given $j,n,m$ if $s=\left\lceil \dfrac{n+m-1-k}{j-1} \right\rceil$, in this paper, we show that the size Ramsey numbers $m_j(P_n,S_{m+1})$ is bounded above by $s$ for $k=\left\lceil \dfrac{n-1}{j} \right\rceil$. Given $j\ge 3$ and $s$, we will obtain an infinite class $(n,m)$ that achieves this upper bound $s$. In the later part of the paper, will also investigate necessary and sufficient conditions needed for the upper bound to hold.</p>

Confines

2019 ◽  
pp. 161-193
Author(s):  
Joseph P. Reidy
Keyword(s):  
Social Relationships ◽  
Vantage Point ◽  
Refugee Camps ◽  
Confined Space ◽  
Simple Characterization ◽  
War Effort

Confined space offers an instructive vantage point into the reconfiguration of social relationships that were central to the emancipation process. In homes and kitchens throughout the slave states, enslaved house servants devised strategies for asserting greater control over their labor and their lives, even when escape to freedom was out of reach. Women and men hired to work in the shops and factories that supported the Confederate war effort interacted with new casts of characters with new possibilities for stretching their customary boundaries and shedding their usual constraints. For freedom-seeking refugees who reached Union lines, refugee camps (generally called "contraband camps") offered shelter and employment, though often under the watchful eyes of proselytizing Northerners. Cities presented special conditions for the breakdown of slavery, as the experience of Washington, D.C., illustrates. The D.C. emancipation act of April 1862 set in motion a contested process that defies the simple characterization of immediate emancipation.

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