scholarly journals An Improvement to Mathon's Cyclotomic Ramsey Colorings

10.37236/239 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Xiaodong Xu ◽  
Stanisław P. Radziszowski
Keyword(s):  
Lower Bound ◽  
Complete Graphs ◽  
Maximum Order ◽  
Ramsey Numbers

In this note we show how to extend Mathon's cyclotomic colorings of the edges of some complete graphs without increasing the maximum order of monochromatic complete subgraphs. This improves the well known lower bound construction for multicolor Ramsey numbers, in particular we obtain $R_3(7) \ge 3214$.

scholarly journals On size multipartite Ramsey numbers for stars

2020 ◽  
Vol 3 (2) ◽  
pp. 109
Author(s):  
Anie Lusiani ◽  
Edy Tri Baskoro ◽  
Suhadi Wido Saputro
Keyword(s):  
Lower Bound ◽  
Disjoint Union ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Ramsey Number ◽  
Complete Graphs ◽  
Ramsey Numbers ◽  
Simple Graphs ◽  
Multipartite Graphs ◽  
Necessary And Sufficient

<p>Burger and Vuuren defined the size multipartite Ramsey number for a pair of complete, balanced, multipartite graphs <em>mj</em>(<em>Ka</em>x<em>b</em>,<em>Kc</em>x<em>d</em>), for natural numbers <em>a,b,c,d</em> and <em>j</em>, where <em>a,c</em> &gt;= 2, in 2004. They have also determined the necessary and sufficient conditions for the existence of size multipartite Ramsey numbers <em>mj</em>(<em>Ka</em>x<em>b</em>,<em>Kc</em>x<em>d</em>). Syafrizal <em>et al</em>. generalized this definition by removing the completeness requirement. For simple graphs <em>G</em> and <em>H</em>, they defined the size multipartite Ramsey number <em>mj</em>(<em>G,H</em>) as the smallest natural number <em>t</em> such that any red-blue coloring on the edges of <em>Kj</em>x<em>t</em> contains a red <em>G</em> or a blue <em>H</em> as a subgraph. In this paper, we determine the necessary and sufficient conditions for the existence of multipartite Ramsey numbers <em>mj</em>(<em>G,H</em>), where both <em>G</em> and <em>H</em> are non complete graphs. Furthermore, we determine the exact values of the size multipartite Ramsey numbers <em>mj</em>(<em>K</em>1,<em>m</em>, <em>K</em>1,<em>n</em>) for all integers <em>m,n &gt;= </em>1 and <em>j </em>= 2,3, where <em>K</em>1,<em>m</em> is a star of order <em>m</em>+1. In addition, we also determine the lower bound of <em>m</em>3(<em>kK</em>1,<em>m</em>, <em>C</em>3), where <em>kK</em>1,<em>m</em> is a disjoint union of <em>k</em> copies of a star <em>K</em>1,<em>m</em> and <em>C</em>3 is a cycle of order 3.</p>

A Lower Bound for Schur Numbers and Multicolor Ramsey Numbers

10.37236/1188 ◽  
1994 ◽  
Vol 1 (1) ◽  
Author(s):  
Geoffrey Exoo
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Ramsey Numbers

For $k \geq 5$, we establish new lower bounds on the Schur numbers $S(k)$ and on the k-color Ramsey numbers of $K_3$.

scholarly journals Minimum Number of Monotone Subsequences of Length 4 in Permutations

2014 ◽  
Vol 24 (4) ◽  
pp. 658-679 ◽  
Author(s):  
JÓZSEF BALOGH ◽  
PING HU ◽  
BERNARD LIDICKÝ ◽  
OLEG PIKHURKO ◽  
BALÁZS UDVARI ◽  
...  
Keyword(s):  
Lower Bound ◽  
Complete Graphs ◽  
Formula Group ◽  
Minimum Number ◽  
Additional Stability ◽  
Edge Colourings ◽  
Image Position

We show that for every sufficiently largen, the number of monotone subsequences of length four in a permutation onnpoints is at least\begin{equation*} \binom{\lfloor{n/3}\rfloor}{4} + \binom{\lfloor{(n+1)/3}\rfloor}{4} + \binom{\lfloor{(n+2)/3}\rfloor}{4}. \end{equation*}Furthermore, we characterize all permutations on [n] that attain this lower bound. The proof uses the flag algebra framework together with some additional stability arguments. This problem is equivalent to some specific type of edge colourings of complete graphs with two colours, where the number of monochromaticK4is minimized. We show that all the extremal colourings must contain monochromaticK4only in one of the two colours. This translates back to permutations, where all the monotone subsequences of length four are all either increasing, or decreasing only.

scholarly journals On the Ramsey Number $R(4,6)$

10.37236/2102 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Geoffrey Exoo
Keyword(s):  
Fixed Point ◽  
Automorphism Group ◽  
Lower Bound ◽  
Heuristic Search ◽  
Ramsey Number ◽  
Complete Graphs ◽  
Search Procedure ◽  
Edge Colorings

The lower bound for the classical Ramsey number $R(4,6)$ is improved from 35 to 36. The author has found 37 new edge colorings of $K_{35}$ that have no complete graphs of order 4 in the first color, and no complete graphs of order 6 in the second color. The most symmetric of the colorings has an automorphism group of order 4, with one fixed point, and is presented in detail. The colorings were found using a heuristic search procedure.

Ramsey Numbers of Some Bipartite Graphs Versus Complete Graphs

2010 ◽  
Vol 27 (1) ◽  
pp. 121-128 ◽  
Author(s):  
Tao Jiang ◽  
Michael Salerno
Keyword(s):  
Bipartite Graphs ◽  
Complete Graphs ◽  
Ramsey Numbers

scholarly journals On the Ramsey numbers for stars versus complete graphs

2010 ◽  
Vol 31 (7) ◽  
pp. 1680-1688 ◽  
Author(s):  
L. Boza ◽  
M. Cera ◽  
P. García-Vázquez ◽  
M.P. Revuelta
Keyword(s):  
Complete Graphs ◽  
Ramsey Numbers

Local Anti-Ramsey Numbers of Graphs

2003 ◽  
Vol 12 (5-6) ◽  
pp. 495-511 ◽  
Author(s):  
Maria Axenovich ◽  
Tao Jiang ◽  
Z Tuza
Keyword(s):  
Complete Graphs ◽  
Ramsey Numbers ◽  
Edge Colourings ◽  
Edge Colouring

A subgraph H in an edge-colouring is properly coloured if incident edges of H are assigned different colours, and H is rainbow if no two edges of H are assigned the same colour. We study properly coloured subgraphs and rainbow subgraphs forced in edge-colourings of complete graphs in which each vertex is incident to a large number of colours.

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