Tanbir Ahmed, Ph. D.
Publications
Papers •
Theses •
Integer Sequences •
Reviews
Papers • Back to top
023. A Note on the Rado numbers \(R_k(3,c)\)
Tanbir Ahmed, Robert Malo, Daniel Schaal, Submitted to a journal.
Ramsey Theory on the Integers
022. On the \(3\)-color off-diagonal generalized Schur numbers \(S(3; 2, k_1, k_2)\)
Tanbir Ahmed, Luis Boza, Maria P. Revuelta, and Maria I. Sanz, Submitted to a journal.
Ramsey Theory on the Integers
SAT-Solving
021. On the \(3\)-color off-diagonal generalized Schur numbers \(S(3; 2,2,k)\)
Tanbir Ahmed, Luis Boza, Maria P. Revuelta, and Maria I. Sanz, Submitted to a journal.
Ramsey Theory on the Integers
SAT-Solving
020. Establishing lower bounds and exact values of the \(2\)-color off-diagonal generalized weak Schur numbers \(WS(2; k_1, k_2)\)
Tanbir Ahmed, Luis Boza, Maria P. Revuelta, and Maria I. Sanz, Submitted to a journal.
Ramsey Theory on the Integers
SAT-Solving
019. Lower bounds and exact values of the \(2\)-color off-diagonal generalized weak Schur numbers \(WS(2; k_1, k_2)\) conference journal
Tanbir Ahmed, Luis Boza, Maria P. Revuelta, and Maria I. Sanz, Procedia Computer Science, 223 (2023), 403-405.
XII LAGOS, 2023.
Ramsey Theory on the Integers
SAT-Solving
MR4742638
018. Exact values and lower bounds on the \(n\)-color weak Schur numbers for \(n = 2, 3\). journal
Tanbir Ahmed, Luis Boza, Maria P. Revuelta, and Maria I. Sanz,
The Ramanujan Journal, 62 (2) (2023), 347-363.
, Open access, (2023).
Ramsey Theory on the Integers
SAT-Solving
MR4642695
017. Thue-Morse substitution and self similar groups and algebras journal
Laurent Bartholdi, José M. R. Caballero, and Tanbir Ahmed, Algebra and Discrete Mathematics, 34 (1) (2022), 1-14.
Self-Similar Algebra
MR4549130
016. A Family of doubly stochastic matrices involving Chebyshev polynomials. journal
Tanbir Ahmed and José M. R. Caballero, Algebra and Discrete Mathematics, 27 (2) (2019), 155-164.
Self-Similar Algebra
MR3982300
015. Power Sum Polynomials as Relaxed EGZ Polynomials. journal
Tanbir Ahmed, Arie Bialostocki, Thang Pham, and Le Anh Vinh,
Integers 19 (2019),
A49.
Zero-Sum Ramsey Theory
MR4017190
014. On Generalized Schur Numbers journal
Tanbir Ahmed and Daniel Schaal,
Experimental Mathematics, 25 (2) (2016), 213-218.
(PDF)
Ramsey Theory on the Integers
SAT-Solving
MR3463569
013. On Colouring Integers without t-AP distance sets journal
Tanbir Ahmed, Algebra and Discrete Mathematics, 22 (1) (2016), 1-10.
(PDF)
Ramsey Theory on the Integers
SAT-Solving
MR3573541
012. On Distance Sets in the Triangular Lattice journal
Tanbir Ahmed and David Wildstrom,
Bull. Inst. Combin. Appl. 75 (2015), 118-127.
(PDF)
Combinatorial Geometry
MR3444619
011. Remembering Hunter Snevily journal
Tanbir Ahmed, André Kézdy, and Douglas B. West,
Bull. Inst. Combin. Appl., 73 (2015), 7-17.
(PDF)
Research Memoir
MR3331369
010. On the van der Waerden numbers \(w(2; 3, t)\) journal
Tanbir Ahmed, Oliver Kullmann, and Hunter Snevily,
Discrete Applied Mathematics, 174 (2014), 27-51.
Ramsey Theory on the Integers
SAT-Solving
MR3215454
--- The paper got a citation in D. E. Knuth's
Volume 4, Fascicle 6: Chapter on Satisfiability
--- See Wikipedia entry of
Van der Waerden numbers for my contributions in that list
--- Recently (2022), Ben Green
in New Lower Bounds for van der Waerden Numbers,
disproved the much-speculated conjecture \(w(2; 3, k)=\cal{O}(k^2)\) (originally formulated by Ron Graham in around 2004, and subsequently supported with data by others and ourselves)
by constructing a red-blue coloring of \(1,2,\ldots,N\) with no blue arithmetic progression of length 3 and no red arithmetic
progression of length \(e^{C(\log N)^{3/4}(\log\log N)^{1/4}}\), and consequently
showing that \(w(2; 3, k)\) is bounded from below by \(k^{b(k)}\) where \(b(k) = c\left({\log{k}\over\log\log{k}}\right)^{1/3}\).
See Erica Klarreich's
Quanta Magazine article
Mathematician Hurls Structure and Disorder Into Century-Old Problem.
009. The α-labeling number of comets is 2 journal
Tanbir Ahmed and Hunter Snevily,
Bull. Inst. Combin. Appl., 72 (2014), 25-40.
(PDF)
Graph Labeling
MR3362514
--- (See Joseph A. Gallian's A Dynamic Survey of Graph Labeling)
008. Sparse Distance Sets in the Triangular Lattice journal
Tanbir Ahmed and Hunter Snevily, Electronic Journal of Combinatorics,
20 (4) (2013), P33.
Combinatorial Geometry
MR3158272
007. Unique Sequences Containing No k-Term Arithmetic Progressions journal
Tanbir Ahmed, Janusz Dybizbański, and Hunter Snevily,
Electronic Journal of Combinatorics,
20 (4) (2013), P29.
Combinatorics
MR3158268
006. Some properties of Roller Coaster Permutations journal
Tanbir Ahmed and Hunter Snevily,
Bull. Inst. Combin. Appl., 68 (2013), 55-69.
(PDF)
Combinatorics
MR3136863
--- (See Biedl et al., Rollercoasters: long sequences without short runs., SIAM J. Discrete Math. 33, No. 2, 845-861 (2019).
--- (See two conjectures
in the Open Problems Garden)
--- (See William Adamczak
A Note on the Structure of Roller Coaster Permutations, preprint, May 2016)
--- (See William Adamczak, Jacob Boni
Roller Coaster Permutations and Partition Numbers, preprint, March 2017)
005. Strict Schur Numbers journal
Tanbir Ahmed, Michael G. Eldredge, Jonathan J. Marler, and Hunter Snevily,
Integers, 13 (2013).
A22.
Ramsey Theory on the Integers
MR3083484
004. Some more van der Waerden numbers journal
Tanbir Ahmed, Journal of Integer Sequences,
16 (2013), Article 13.4.4.
Ramsey Theory on the Integers
SAT-Solving
MR3056628
003. On computation of exact van der Waerden numbers journal
Tanbir Ahmed, Integers, 12 (3) (2012), 417-425.
Online version: 11 (2011), A71.
Ramsey Theory on the Integers
SAT-Solving
MR2955523
002. Two new van der Waerden numbers: w(2; 3, 17) and w(2; 3, 18) journal
Tanbir Ahmed, Integers,
10 (2010), 369-377, A32.
Ramsey Theory on the Integers
SAT-Solving
MR2684128
-- First applied DIVIDE-AND-CONQUER Distributed SAT-Solving to compute these numbers in around 2009.
001. Some new van der Waerden numbers and some van der Waerden-type numbers journal
Tanbir Ahmed, Integers,
9 (2009), 65-76, A06
Ramsey Theory on the Integers
SAT-Solving
MR2506138
--- (See tawSolver 1.0:
an efficient implementation of the DPLL Algorithm.)
Theses • Back to top
Some Results in Extremal Combinatorics
[PDF]
Ph.D. Thesis, 2013, Concordia University.
Advisor: Prof. Clement Lam
An Implementation of the DPLL Algorithm
[PDF]
M. Comp. Sci. Thesis, 2009, Concordia University.
Advisor: Prof. Vašek Chvátal
Integer Sequences • Back to top
Van der Waerden numbers \(w(2; 3, n)\) for \(n \geqslant 3\). Only values for \(n=3,4,\ldots,19\) are known
A171081: 9, 18, 22, 32, 46, 58, 77, 97, 114, 135, 160, 186, 218, 238, 279, 312, 349
Van der Waerden numbers \(w(2; 4, n)\) for \(n \geqslant 4\). Only values for \(n=4,5,\ldots,9\) are known
A007784: 35, 55, 73, 109, 146, 309
Van der Waerden numbers \(w(2; 5, n)\) for \(n \geqslant 5\). Only values for \(n=5,6,7\) are known
A217037: 178, 206, 260
Van der Waerden numbers \(w(3; 3, 3, n)\) for \(n \geqslant 3\). Only values for \(n=3,4,5,6\) are known
A217235: 27, 51, 80, 107
Van der Waerden numbers \(w(j+2; t_0,t_1,...,t_{j-1}, 3, 3)\) with \(t_0 = t_1 = ... = t_{j-1} = 2\).
A217005: 9, 14, 17, 20, 21, 24, 25, 28, 31, 33, 35, 37,
39, 42, 44, 46, 48, 50, 51
Van der Waerden numbers \(w(j+2; t_0,t_1,...,t_{j-1}, 3, 4)\) with \(t_0 = t_1 = ... = t_{j-1} = 2\).
A217058: 18, 21, 25, 29, 33, 36, 40, 42, 45, 48, 52, 55
Van der Waerden numbers \(w(j+2; t_0,t_1,...,t_{j-1}, 3, 5)\) with \(t_0 = t_1 = ... = t_{j-1} = 2\).
A217059: 22, 32, 43, 44, 50, 55, 61, 65, 70
Van der Waerden numbers \(w(j+2; t_0,t_1,...,t_{j-1}, 3, 6)\) with \(t_0 = t_1 = ... = t_{j-1} = 2\).
A217060: 32, 40, 48, 56, 60, 65, 71
Van der Waerden numbers \(w(j+2; t_0,t_1,...,t_{j-1}, 4, 4)\) with \(t_0 = t_1 = ... = t_{j-1} = 2\).
A217007: 35, 40, 53, 54, 56, 66, 67
Van der Waerden numbers \(w(j+2; t_0,t_1,...,t_{j-1}, 4, 5)\) with \(t_0 = t_1 = ... = t_{j-1} = 2\).
A217236: 55, 71, 75, 79
Van der Waerden numbers \(w(j+2; t_0,t_1,...,t_{j-1}, 4, 6)\) with \(t_0 = t_1 = ... = t_{j-1} = 2\).
A217237: 73, 83, 93, 101
Van der Waerden numbers \(w(j+3; t_0,t_1,...,t_{j-1}, 3, 3, 3)\) with \(t_0 = t_1 = ... = t_{j-1} = 2\).
A217008: 27, 40, 41, 42, 45, 49, 52
Reviews • Back to top
In Math. Reviews
MR3274878,
MR3231981,
MR3194752,
MR3190574,
MR3134264,
MR3063975,
MR3062784,
MR3061011
In Computing Reviews
CR141762,
CR141855,
CR142155,
CR142308,
CR142738,
CR142894
Referee services
American Mathematical Society (AMS) Mathematics of Computation •
Bulletin of the Korean Mathematical Society (KMS) •
Discrete Applied Mathematics •
Integers •
Rocky Mountain Journal of Mathematics •
The Ramanujan Journal
Papers • Theses • Integer Sequences • Reviews
Papers • Back to top
Tanbir Ahmed, Robert Malo, Daniel Schaal, Submitted to a journal.
Ramsey Theory on the Integers
022. On the \(3\)-color off-diagonal generalized Schur numbers \(S(3; 2, k_1, k_2)\)
Tanbir Ahmed, Luis Boza, Maria P. Revuelta, and Maria I. Sanz, Submitted to a journal.
Ramsey Theory on the Integers SAT-Solving
021. On the \(3\)-color off-diagonal generalized Schur numbers \(S(3; 2,2,k)\)
Tanbir Ahmed, Luis Boza, Maria P. Revuelta, and Maria I. Sanz, Submitted to a journal.
Ramsey Theory on the Integers SAT-Solving
020. Establishing lower bounds and exact values of the \(2\)-color off-diagonal generalized weak Schur numbers \(WS(2; k_1, k_2)\)
Tanbir Ahmed, Luis Boza, Maria P. Revuelta, and Maria I. Sanz, Submitted to a journal.
Ramsey Theory on the Integers SAT-Solving
019. Lower bounds and exact values of the \(2\)-color off-diagonal generalized weak Schur numbers \(WS(2; k_1, k_2)\) conference journal
Tanbir Ahmed, Luis Boza, Maria P. Revuelta, and Maria I. Sanz, Procedia Computer Science, 223 (2023), 403-405. XII LAGOS, 2023.
Ramsey Theory on the Integers SAT-Solving MR4742638
018. Exact values and lower bounds on the \(n\)-color weak Schur numbers for \(n = 2, 3\). journal
Tanbir Ahmed, Luis Boza, Maria P. Revuelta, and Maria I. Sanz, The Ramanujan Journal, 62 (2) (2023), 347-363. , Open access, (2023).
Ramsey Theory on the Integers SAT-Solving MR4642695
017. Thue-Morse substitution and self similar groups and algebras journal
Laurent Bartholdi, José M. R. Caballero, and Tanbir Ahmed, Algebra and Discrete Mathematics, 34 (1) (2022), 1-14.
Self-Similar Algebra MR4549130
016. A Family of doubly stochastic matrices involving Chebyshev polynomials. journal
Tanbir Ahmed and José M. R. Caballero, Algebra and Discrete Mathematics, 27 (2) (2019), 155-164.
Self-Similar Algebra MR3982300
015. Power Sum Polynomials as Relaxed EGZ Polynomials. journal
Tanbir Ahmed, Arie Bialostocki, Thang Pham, and Le Anh Vinh, Integers 19 (2019), A49.
Zero-Sum Ramsey Theory MR4017190
014. On Generalized Schur Numbers journal
Tanbir Ahmed and Daniel Schaal, Experimental Mathematics, 25 (2) (2016), 213-218. (PDF)
Ramsey Theory on the Integers SAT-Solving MR3463569
013. On Colouring Integers without t-AP distance sets journal
Tanbir Ahmed, Algebra and Discrete Mathematics, 22 (1) (2016), 1-10. (PDF)
Ramsey Theory on the Integers SAT-Solving MR3573541
012. On Distance Sets in the Triangular Lattice journal
Tanbir Ahmed and David Wildstrom, Bull. Inst. Combin. Appl. 75 (2015), 118-127. (PDF)
Combinatorial Geometry MR3444619
011. Remembering Hunter Snevily journal
Tanbir Ahmed, André Kézdy, and Douglas B. West, Bull. Inst. Combin. Appl., 73 (2015), 7-17. (PDF)
Research Memoir MR3331369
010. On the van der Waerden numbers \(w(2; 3, t)\) journal
Tanbir Ahmed, Oliver Kullmann, and Hunter Snevily, Discrete Applied Mathematics, 174 (2014), 27-51.
Ramsey Theory on the Integers SAT-Solving MR3215454
--- The paper got a citation in D. E. Knuth's Volume 4, Fascicle 6: Chapter on Satisfiability
--- See Wikipedia entry of Van der Waerden numbers for my contributions in that list
--- Recently (2022), Ben Green in New Lower Bounds for van der Waerden Numbers, disproved the much-speculated conjecture \(w(2; 3, k)=\cal{O}(k^2)\) (originally formulated by Ron Graham in around 2004, and subsequently supported with data by others and ourselves) by constructing a red-blue coloring of \(1,2,\ldots,N\) with no blue arithmetic progression of length 3 and no red arithmetic progression of length \(e^{C(\log N)^{3/4}(\log\log N)^{1/4}}\), and consequently showing that \(w(2; 3, k)\) is bounded from below by \(k^{b(k)}\) where \(b(k) = c\left({\log{k}\over\log\log{k}}\right)^{1/3}\). See Erica Klarreich's Quanta Magazine article Mathematician Hurls Structure and Disorder Into Century-Old Problem.
009. The α-labeling number of comets is 2 journal
Tanbir Ahmed and Hunter Snevily, Bull. Inst. Combin. Appl., 72 (2014), 25-40. (PDF)
Graph Labeling MR3362514
--- (See Joseph A. Gallian's A Dynamic Survey of Graph Labeling)
008. Sparse Distance Sets in the Triangular Lattice journal
Tanbir Ahmed and Hunter Snevily, Electronic Journal of Combinatorics, 20 (4) (2013), P33.
Combinatorial Geometry MR3158272
007. Unique Sequences Containing No k-Term Arithmetic Progressions journal
Tanbir Ahmed, Janusz Dybizbański, and Hunter Snevily, Electronic Journal of Combinatorics, 20 (4) (2013), P29.
Combinatorics MR3158268
006. Some properties of Roller Coaster Permutations journal
Tanbir Ahmed and Hunter Snevily, Bull. Inst. Combin. Appl., 68 (2013), 55-69. (PDF)
Combinatorics MR3136863
--- (See Biedl et al., Rollercoasters: long sequences without short runs., SIAM J. Discrete Math. 33, No. 2, 845-861 (2019).
--- (See two conjectures in the Open Problems Garden)
--- (See William Adamczak A Note on the Structure of Roller Coaster Permutations, preprint, May 2016)
--- (See William Adamczak, Jacob Boni Roller Coaster Permutations and Partition Numbers, preprint, March 2017)
005. Strict Schur Numbers journal
Tanbir Ahmed, Michael G. Eldredge, Jonathan J. Marler, and Hunter Snevily, Integers, 13 (2013). A22.
Ramsey Theory on the Integers MR3083484
004. Some more van der Waerden numbers journal
Tanbir Ahmed, Journal of Integer Sequences, 16 (2013), Article 13.4.4.
Ramsey Theory on the Integers SAT-Solving MR3056628
003. On computation of exact van der Waerden numbers journal
Tanbir Ahmed, Integers, 12 (3) (2012), 417-425. Online version: 11 (2011), A71.
Ramsey Theory on the Integers SAT-Solving MR2955523
002. Two new van der Waerden numbers: w(2; 3, 17) and w(2; 3, 18) journal
Tanbir Ahmed, Integers, 10 (2010), 369-377, A32.
Ramsey Theory on the Integers SAT-Solving MR2684128
-- First applied DIVIDE-AND-CONQUER Distributed SAT-Solving to compute these numbers in around 2009.
001. Some new van der Waerden numbers and some van der Waerden-type numbers journal
Tanbir Ahmed, Integers, 9 (2009), 65-76, A06
Ramsey Theory on the Integers SAT-Solving MR2506138
--- (See tawSolver 1.0: an efficient implementation of the DPLL Algorithm.)
Theses • Back to top
Some Results in Extremal Combinatorics [PDF]Ph.D. Thesis, 2013, Concordia University.
Advisor: Prof. Clement Lam
An Implementation of the DPLL Algorithm [PDF]
M. Comp. Sci. Thesis, 2009, Concordia University.
Advisor: Prof. Vašek Chvátal
Integer Sequences • Back to top
Van der Waerden numbers \(w(2; 3, n)\) for \(n \geqslant 3\). Only values for \(n=3,4,\ldots,19\) are knownA171081: 9, 18, 22, 32, 46, 58, 77, 97, 114, 135, 160, 186, 218, 238, 279, 312, 349
Van der Waerden numbers \(w(2; 4, n)\) for \(n \geqslant 4\). Only values for \(n=4,5,\ldots,9\) are known
A007784: 35, 55, 73, 109, 146, 309
Van der Waerden numbers \(w(2; 5, n)\) for \(n \geqslant 5\). Only values for \(n=5,6,7\) are known
A217037: 178, 206, 260
Van der Waerden numbers \(w(3; 3, 3, n)\) for \(n \geqslant 3\). Only values for \(n=3,4,5,6\) are known
A217235: 27, 51, 80, 107
Van der Waerden numbers \(w(j+2; t_0,t_1,...,t_{j-1}, 3, 3)\) with \(t_0 = t_1 = ... = t_{j-1} = 2\).
A217005: 9, 14, 17, 20, 21, 24, 25, 28, 31, 33, 35, 37, 39, 42, 44, 46, 48, 50, 51
Van der Waerden numbers \(w(j+2; t_0,t_1,...,t_{j-1}, 3, 4)\) with \(t_0 = t_1 = ... = t_{j-1} = 2\).
A217058: 18, 21, 25, 29, 33, 36, 40, 42, 45, 48, 52, 55
Van der Waerden numbers \(w(j+2; t_0,t_1,...,t_{j-1}, 3, 5)\) with \(t_0 = t_1 = ... = t_{j-1} = 2\).
A217059: 22, 32, 43, 44, 50, 55, 61, 65, 70
Van der Waerden numbers \(w(j+2; t_0,t_1,...,t_{j-1}, 3, 6)\) with \(t_0 = t_1 = ... = t_{j-1} = 2\).
A217060: 32, 40, 48, 56, 60, 65, 71
Van der Waerden numbers \(w(j+2; t_0,t_1,...,t_{j-1}, 4, 4)\) with \(t_0 = t_1 = ... = t_{j-1} = 2\).
A217007: 35, 40, 53, 54, 56, 66, 67
Van der Waerden numbers \(w(j+2; t_0,t_1,...,t_{j-1}, 4, 5)\) with \(t_0 = t_1 = ... = t_{j-1} = 2\).
A217236: 55, 71, 75, 79
Van der Waerden numbers \(w(j+2; t_0,t_1,...,t_{j-1}, 4, 6)\) with \(t_0 = t_1 = ... = t_{j-1} = 2\).
A217237: 73, 83, 93, 101
Van der Waerden numbers \(w(j+3; t_0,t_1,...,t_{j-1}, 3, 3, 3)\) with \(t_0 = t_1 = ... = t_{j-1} = 2\).
A217008: 27, 40, 41, 42, 45, 49, 52
Reviews • Back to top
In Math. ReviewsMR3274878, MR3231981, MR3194752, MR3190574, MR3134264, MR3063975, MR3062784, MR3061011
In Computing Reviews
CR141762, CR141855, CR142155, CR142308, CR142738, CR142894
Referee services
American Mathematical Society (AMS) Mathematics of Computation • Bulletin of the Korean Mathematical Society (KMS) • Discrete Applied Mathematics • Integers • Rocky Mountain Journal of Mathematics • The Ramanujan Journal