An improved lower bound for multicolor Ramsey numbers and a problem of Erdős

2022 ◽  
Vol 188 ◽  
pp. 105579
Author(s):  
Will Sawin
Keyword(s):  
Lower Bound ◽  
Ramsey Numbers

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 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 A lower bound for irredundant Ramsey numbers

1998 ◽  
Vol 183 (1-3) ◽  
pp. 185-192 ◽  
Author(s):  
Michael Krivelevich
Keyword(s):  
Lower Bound ◽  
Ramsey Numbers

scholarly journals New Lower Bound Formulas for Multicolored Ramsey Numbers

10.37236/1630 ◽  
2002 ◽  
Vol 9 (1) ◽  
Author(s):  
Aaron Robertson
Keyword(s):  
Lower Bound ◽  
Ramsey Numbers

We give two lower bound formulas for multicolored Ramsey numbers. These formulas improve the bounds for several small multicolored Ramsey numbers.

scholarly journals Off-Diagonal Ordered Ramsey Numbers of Matchings

10.37236/8085 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Dhruv Rohatgi
Keyword(s):  
Lower Bound ◽  
Chromatic Number ◽  
Upper Bound ◽  
Edge Coloring ◽  
Ramsey Number ◽  
Relative Order ◽  
Ramsey Numbers ◽  
Asymptotic Bounds ◽  
Special Cases ◽  
Ordered Graphs

For ordered graphs $G$ and $H$, the ordered Ramsey number $r_<(G,H)$ is the smallest $n$ such that every red/blue edge coloring of the complete ordered graph on vertices $\{1,\dots,n\}$ contains either a blue copy of $G$ or a red copy of $H$, where the embedding must preserve the relative order of vertices. One number of interest, first studied by Conlon, Fox, Lee, and Sudakov, is the off-diagonal ordered Ramsey number $r_<(M, K_3)$, where $M$ is an ordered matching on $n$ vertices. In particular, Conlon et al. asked what asymptotic bounds (in $n$) can be obtained for $\max r_<(M, K_3)$, where the maximum is over all ordered matchings $M$ on $n$ vertices. The best-known upper bound is $O(n^2/\log n)$, whereas the best-known lower bound is $\Omega((n/\log n)^{4/3})$, and Conlon et al. hypothesize that there is some fixed $\epsilon > 0$ such that $r_<(M, K_3) = O(n^{2-\epsilon})$ for every ordered matching $M$. We resolve two special cases of this conjecture. We show that the off-diagonal ordered Ramsey numbers for ordered matchings in which edges do not cross are nearly linear. We also prove a truly sub-quadratic upper bound for random ordered matchings with interval chromatic number $2$.

scholarly journals Batas Bawah Bilangan Ramsey pada Graf Bintang S_10 Versus Roda W_12 Lower Bound of the Ramsey Number for the Star Graph S_10 Versus Wheels W_12

2018 ◽  
Vol 3 (1) ◽  
pp. 471
Author(s):  
Hamdana Hadaming ◽  
Andi Ardhila Wahyudi
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Lower Boundary ◽  
Ramsey Number ◽  
Star Graph ◽  
Point Graph ◽  
Ramsey Numbers ◽  
Critical Graph ◽  
Star Versus

Bilangan Ramsey untuk graf  terhadap graf , dinotasikan dengan  adalah bilangan bulat terkecil  sedemikian sehingga untuk setiap graf  dengan orde akan memenuhi sifat berikut:  memuat graf  atau komplemen dari  memuat graf .Penelitian ini bertujuan untuk  menentukan graf kritis maksimum  dan  dengan genap. Berdasarkan batas bawah tersebut di tentukan batas atas minimum sehingga diperoleh nilai bilangan Ramsey untuk graf bintang  versus , atau . Dengan demikian penentuan batas bawah bilangan Ramsey  dilakukan dengan cara batas bawah yang  diberikan oleh Chavatal dan Harary, untuk bilangan Ramsey pada graf bintang  versus  adalah , dengan  adalah bilangan kromatik titik graf roda  dan  adalah kardinalitas komponen terbesar graf . Berdasarkan batas bawah Chavatal dan Harary tersebut dikonstruksi graf kritis untuk  dan  yang ordenya lebih besar dari nilai batas bawah yang diberikan Chavatal dan Harary. Orde dari graf kritis tersebut merupakan batas bawah terbaik untuk . Kata kunci: Bilangan Ramsey, bintang, roda AbstractRamsey Numbers for a graph  to a graph , denoted by   is the smallest integer n such that for every graph  of order  either  the following meeet:  contains a graph  or the complement of  contains the graph . This aims of the study to determine the maximum critical graph  and . Based on the lower bound of the specified minimum upper bound in order to obtain numerical values for the Ramsey graph  Star versus  , or . Thus the determination of Ramsey numbers .  is done by determine the lower boundary and upper bound. The lower bound given by Chavatal and Harary, for ramsey number for star graph versus wheel   is , is a point graph of chromatic number wheel  and  is the cardinality of the largest component of the graph . Based on the lower bound Chavatal and Harary graph is constructed critical to  and  are poin greater than the lower bound value given Chavatal and Harary. Order of the critical graph is the best lower bound for . Keywords : Ramsey number, Stars, and Wheels

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>

Lower Bounds for the Ramsey Numbers

1970 ◽  
Vol 13 (2) ◽  
pp. 227-229 ◽  
Author(s):  
Pierre Robillard
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Ramsey Numbers ◽  
Geometrical Argument

Abstract.A lower bound for a family of Ramsey numbers is derived using a geometrical argument.

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