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A Lower Bound for Schur Numbers and Multicolor Ramsey Numbers

Geoffrey Exoo
1994 Electronic Journal of Combinatorics  
For $k \geq 5$, we establish new lower bounds on the Schur numbers $S(k)$ and on the k-color Ramsey numbers of $K_3$.  ...  For Ramsey numbers we know R 2 (3) = 6 and R 3 (3) = 17; the current bounds on R 4 (3) are 51 and 65 [5] .  ...  The best previously published bounds for S(5) are 157 ≤ S(5) ≤ 321, the lower bound was proved in [4] and the upper bound in [6] .  ... 
doi:10.37236/1188 fatcat:wlzfm7t76ngfle2aq4lpbkqdvu

Difference Ramsey Numbers and Issai Numbers

Aaron Robertson
2000 Advances in Applied Mathematics  
We present a recursive algorithm for finding good lower bounds for the classical Ramsey numbers.  ...  Using notions from this algorithm we then give some results for a generalization of the Generalized Schur numbers, which we call Issai numbers.  ...  Acknowledgments I would like to thank my advisor, Doron Zeilberger, for his guidance, his support, and for sharing his mathematical philosophies.  ... 
doi:10.1006/aama.2000.0678 fatcat:layjljbeufaihfqqfahfw3ui6a

Difference Ramsey Numbers and Issai Numbers [article]

Aaron Robertson
1999 arXiv   pre-print
We present a recursive algorithm for finding good lower bounds for the classical Ramsey numbers.  ...  Using notions from this algorithm we then give some results for generalized Schur numbers, which we call Issai numbers.  ...  Acknowledgment I would like to thank my advisor, Doron Zeilberger, for his guidence, his support, and for sharing his mathematical philosophies.  ... 
arXiv:math/9904023v1 fatcat:riw7vcjygvb4nmau54qppszapm

Constructive Lower Bounds on Classical Multicolor Ramsey Numbers

Xu Xiaodong, Xie Zheng, Geoffrey Exoo, Stanisław P. Radziszowski
2004 Electronic Journal of Combinatorics  
This paper studies lower bounds for classical multicolor Ramsey numbers, first by giving a short overview of past results, and then by presenting several general constructions establishing new lower bounds  ...  for many diagonal and off-diagonal multicolor Ramsey numbers.  ...  Similarly, the recurrence in Theorem 1 below can be applied to derive lower bounds for multicolor Ramsey numbers.  ... 
doi:10.37236/1788 fatcat:zb6pnmtuavasxbuc7v66fsumji

On a Diagonal Conjecture for Classical Ramsey Numbers [article]

Meilian Liang, Stanisław Radziszowski, Xiaodong Xu
2019 arXiv   pre-print
., k_r) denote the classical r-color Ramsey number for integers k_i > 2.  ...  We obtain some implications of this conjecture, present evidence for its validity, and discuss related problems. Let R_r(k) stand for the r-color Ramsey number R(k, ..., k).  ...  Known lower bounds L 1 and L 2 on Ramsey numbers R(a 1 , · · · , a r−2 , a r−1 − 1, a r + 1) and R(a 1 , · · · , a r ), for some DC-adjacent pairs A 1 = a 1 , · · · , a r−2 , a r−1 − 1, a r + 1 and A 2  ... 
arXiv:1810.11386v2 fatcat:tr352ks2i5bc3ddm3cd5t6adnq

Symmetric Sum-Free Partitions and Lower Bounds for Schur Numbers

Harold Fredricksen, Melvin M. Sweet
2000 Electronic Journal of Combinatorics  
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 $R_7(3)$.  ...  Using (2) we obtain the following lower bounds for the Ramsey numbers R 6 (3) and R 7 (3): R 6 (3) ≥ 538 R 7 (3) ≥1682. This improves the bounds given in Radziszowski's survey paper.  ...  For earlier work on lower bounds for S(k) see H. Fredricksen [5] and A. Beutelspacher and W. Brestovansky [2] . The best previously known lower bound S(6) ≥ 481 for S (6) follows from (1) .  ... 
doi:10.37236/1510 fatcat:adcv2f4q7bgc7b4anxfdbbicqe

New lower bounds for Schur and weak Schur numbers [article]

Romain Ageron, Paul Casteras, Thibaut Pellerin, Yann Portella, Arpad Rimmel, Joanna Tomasik
2022 arXiv   pre-print
Finding new templates leads to explicit partitions improving lower bounds as well as the growth rate for Schur numbers, weak Schur numbers, and multicolor Ramsey numbers R_n(3).  ...  This article provides new lower bounds for both Schur and weak Schur numbers by exploiting a "template"-based approach. The concept of "template" is also generalized to weak Schur numbers.  ...  Last but not least, we would like to thank Marijn Heule for providing the backdoors used in the computation of the new lower bound for WS (6) .  ... 
arXiv:2112.03175v2 fatcat:t5ctssxgfbbmffhijytketh2cq

Multicolor list Ramsey numbers grow exponentially [article]

Jacob Fox, Xiaoyu He, Sammy Luo, Max Wenqiang Xu
2022 arXiv   pre-print
The list Ramsey number R_ℓ(H,k), recently introduced by Alon, Bucić, Kalvari, Kuperwasser, and Szabó, is a list-coloring variant of the classical Ramsey number.  ...  They showed that if H is a fixed r-uniform hypergraph that is not r-partite and the number of colors k goes to infinity, e^Ω(√(k))≤ R_ℓ (H,k) ≤ e^O(k).  ...  The authors are grateful to József Balogh, Matija Bucić and Yuval Wigderson for helpful conversations, and the anonymous referees for helpful comments.  ... 
arXiv:2103.15175v2 fatcat:bxzrj7klqnfqhlfe4z43swsxj4

Page 8425 of Mathematical Reviews Vol. , Issue 2000m [page]

2000 Mathematical Reviews  
Summary: “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¢(3) and R7(3).  ...  Symmetric sum-free partitions and lower bounds for Schur numbers. (English summary) Electron. J. Combin. 7 (2000), no. 1, Research Paper 32, 9 pp. (electronic).  ... 

Weak Schur numbers and the search for G.W. Walker's lost partitions

S. Eliahou, J.M. Marín, M.P. Revuelta, M.I. Sanz
2012 Computers and Mathematics with Applications  
With an analogous construction for k = 6, we obtain WS(6) ≥ 572.  ...  Given k ≥ 1, let WS(k) denote the largest integer n for which {1, . . . , n} admits a partition into k weakly sum-free subsets. In 1952, G.W. Walker claimed the value WS(5) = 196, without proof.  ...  We have further improved the lower bound on WS(6), which is now given by WS(6) ≥ 575.  ... 
doi:10.1016/j.camwa.2011.11.006 fatcat:2inailyxzba4takjblolnnkoxy

Small Ramsey Numbers

Stanisław Radziszowski
2011 Electronic Journal of Combinatorics  
We present data which, to the best of our knowledge, includes all known nontrivial values and bounds for specific graph, hypergraph and multicolor Ramsey numbers, where the avoided graphs are complete  ...  We give references to all cited bounds and values, as well as to previous similar compilations.  ...  . , k r )−coloring, which can be considered a special case of Schur partitions defining (symmetric) Schur numbers. Many lower bounds for Ramsey numbers were established by cyclic colorings.  ... 
doi:10.37236/21 fatcat:xsh3n7wd2fbibgxla3okg5dfxq

Ramsey Theory Applications

Vera Rosta
2004 Electronic Journal of Combinatorics  
There are many interesting applications of Ramsey theory, these include results in number theory, algebra, geometry, topology, set theory, logic, ergodic theory, information theory and theoretical computer  ...  Relations of Ramsey-type theorems to various fields in mathematics are well documented in published books and monographs.  ...  Acknowledgment I am grateful to Noga Alon for encouragements and fruitful suggestions. I also thank Gyula Károlyi for helpful comments.  ... 
doi:10.37236/34 fatcat:gxrfo23hzzewjg7rez76d4xx4i

Bounded VC-Dimension Implies the Schur-Erdős Conjecture

Jacob Fox, János Pach, Andrew Suk, Danny Z. Chen, Sergio Cabello
2020 International Symposium on Computational Geometry  
In 1916, Schur introduced the Ramsey number r(3;m), which is the minimum integer n > 1 such that for any m-coloring of the edges of the complete graph K_n, there is a monochromatic copy of K₃.  ...  , and a simple construction demonstrates that r(3;m) ≥ 2^Ω(m). An old conjecture of Erdős states that r(3;m) = 2^Θ(m).  ...  Concluding remarks We have established tight bounds for multicolor Ramsey numbers for graphs with bounded VCdimension.  ... 
doi:10.4230/lipics.socg.2020.46 dblp:conf/compgeom/FoxPS20 fatcat:unasj3albbhbvar3e4df56dguy

Bounded VC-dimension implies the Schur-Erdos conjecture [article]

Jacob Fox, Janos Pach, Andrew Suk
2019 arXiv   pre-print
In 1916, Schur introduced the Ramsey number r(3;m), which is the minimum integer n such that for any m-coloring of the edges of the complete graph K_n, there is a monochromatic copy of K_3.  ...  , and a simple construction demonstrates that r(3;m) ≥ 2^Ω(m). An old conjecture of Erdős states that r(3;m) = 2^Θ(m).  ...  Concluding remarks We have established tight bounds for multicolor Ramsey numbers for graphs with bounded VCdimension.  ... 
arXiv:1912.02342v1 fatcat:o7sou3wzh5ardnz3n6cx4tjoga

Semi-algebraic colorings of complete graphs [article]

Jacob Fox, Janos Pach, Andrew Suk
2018 arXiv   pre-print
., and on a Szemerédi-type regularity lemma for multicolored semi-algebraic graphs, which is of independent interest.  ...  For p> 3 and m> 2, the classical Ramsey number R(p;m) is the smallest positive integer n such that any m-coloring of the edges of K_n, the complete graph on n vertices, contains a monochromatic K_p.  ...  It is a major open problem in Ramsey theory, for which Erdős offered some price money, to close the gap between the lower and upper bounds for R(3; m).  ... 
arXiv:1505.07429v2 fatcat:ny4zrpytjbhuffba3mdbznjaqq
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