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A Lower Bound for Schur Numbers and Multicolor Ramsey Numbers
1994
Electronic Journal of Combinatorics
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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
2000
Advances in Applied Mathematics
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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]
1999
arXiv
pre-print
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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
2004
Electronic Journal of Combinatorics
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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]
2019
arXiv
pre-print
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2018 pre-print arXiv:1810.11386v1 fatcat:uknqtjqti5er3cv3vwoq4e7hgi
., 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
2000
Electronic Journal of Combinatorics
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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]
2022
arXiv
pre-print
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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]
2022
arXiv
pre-print
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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
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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
2012
Computers and Mathematics with Applications
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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
2011
Electronic Journal of Combinatorics
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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
2004
Electronic Journal of Combinatorics
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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
2020
International Symposium on Computational Geometry
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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]
2019
arXiv
pre-print
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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]
2018
arXiv
pre-print
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., 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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