scholarly journals Symmetric Sum-Free Partitions and Lower Bounds for Schur Numbers

10.37236/1510 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
Harold Fredricksen ◽  
Melvin M. Sweet
Keyword(s):  
Lower Bounds ◽  
Ramsey Numbers

We give new lower bounds for the Schur numbers $S(6)$ and $S(7)$. This will imply new lower bounds for the Multicolor Ramsey Numbers $R_6(3)$ and $R_7(3)$. We also make several observations concerning symmetric sum-free partitions into 5 sets.

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 New lower bounds for seven classical Ramsey numbers R(3,q)

2009 ◽  
Vol 22 (3) ◽  
pp. 365-368 ◽  
Author(s):  
Kang Wu ◽  
Wenlong Su ◽  
Haipeng Luo ◽  
Xiaodong Xu
Keyword(s):  
Lower Bounds ◽  
Ramsey Numbers

An Algorithm for Finding Optimal Lower Bounds on Ramsey Numbers Based on Cyclic Graphs

2012 ◽  
Vol 9 (10) ◽  
pp. 1603-1605 ◽  
Author(s):  
Fei Deng ◽  
Zehui Shao ◽  
Xiaodong Xu
Keyword(s):  
Lower Bounds ◽  
Ramsey Numbers ◽  
Cyclic Graphs

New lower bounds of some diagonal Ramsey numbers

1983 ◽  
Vol 7 (1) ◽  
pp. 149-151 ◽  
Author(s):  
Filip Guldan ◽  
Pavel Tomasta
Keyword(s):  
Lower Bounds ◽  
Ramsey Numbers

scholarly journals More Constructive Lower Bounds on Classical Ramsey Numbers

2011 ◽  
Vol 25 (1) ◽  
pp. 394-400 ◽  
Author(s):  
Xiaodong Xu ◽  
Zehui Shao ◽  
StanisŁaw P. Radziszowski
Keyword(s):  
Lower Bounds ◽  
Ramsey Numbers

scholarly journals New lower bounds for classical Ramsey numbers R(5,13) and R(5,14)

1999 ◽  
Vol 12 (6) ◽  
pp. 121-122 ◽  
Author(s):  
Wenlong Su ◽  
Haipeng Luo ◽  
Y.-Q. Shen
Keyword(s):  
Lower Bounds ◽  
Ramsey Numbers

scholarly journals A Generalization of Generalized Paley Graphs and New Lower Bounds for $R(3,q)$

10.37236/474 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Kang Wu ◽  
Wenlong Su ◽  
Haipeng Luo ◽  
Xiaodong Xu
Keyword(s):  
Finite Fields ◽  
Lower Bounds ◽  
Independent Set ◽  
Maximum Independent Set ◽  
Ramsey Numbers ◽  
Cyclic Graphs ◽  
Paley Graphs

Generalized Paley graphs are cyclic graphs constructed from quadratic or higher residues of finite fields. Using this type of cyclic graphs to study the lower bounds for classical Ramsey numbers, has high computing efficiency in both looking for parameter sets and computing clique numbers. We have found a new generalization of generalized Paley graphs, i.e. automorphism cyclic graphs, also having the same advantages. In this paper we study the properties of the parameter sets of automorphism cyclic graphs, and develop an algorithm to compute the order of the maximum independent set, based on which we get new lower bounds for $8$ classical Ramsey numbers: $R(3,22) \geq 131$, $R(3,23) \geq 137$, $R(3,25) \geq 154$, $R(3,28) \geq 173$, $R(3,29) \geq 184$, $R(3,30) \geq 190$, $R(3,31) \geq 199$, $R(3,32) \geq 214$. Furthermore, we also get $R(5,23) \geq 521$ based on $R(3,22) \geq 131$. These nine results above improve their corresponding best known lower bounds.

scholarly journals Applying Tabu Search to Determine New Ramsey Graphs

10.37236/1230 ◽  
1996 ◽  
Vol 3 (1) ◽  
Author(s):  
Konrad Piwakowski
Keyword(s):  
Tabu Search ◽  
Lower Bounds ◽  
Search Algorithm ◽  
Tabu Search Algorithm ◽  
Ramsey Numbers ◽  
Ramsey Graphs

In this note an adaptation of heuristic tabu search algorithm for finding Ramsey graphs is presented. As a result, seven new lower bounds for classical Ramsey numbers are established: $R(3,13)\geq 59$, $R(4,10)\geq 80$, $R(4,11)\geq 96$, $R(4,12)\geq 106$, $R(4,13)\geq 118$, $R(4,14)\geq 129$, and $R(5,8)\geq 95$.

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