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Ramsey Numbers for Matroids

Talmage James Reid
1997 European journal of combinatorics (Print)  
It is natural then to construct Ramsey numbers for matroids replacing the complete and independent subgraphs considered in classical Ramsey theory by circuits and cocircuits of matroids .  ...  In order to do this , we provide analogues in the matroid Ramsey numbers to classical graph theorems on Ramsey numbers .  ...  It is natural then to construct Ramsey numbers for matroids replacing the complete and independent subgraphs considered in classical Ramsey theory by circuits and cocircuits of matroids .  ... 
doi:10.1006/eujc.1995.0112 fatcat:ojb25uzevvgstnxglh5xpq4amq

Page 7325 of Mathematical Reviews Vol. , Issue 97M [page]

1997 Mathematical Reviews  
{For the entire collection see MR 97i:05002.} 97m:05067 05B35 0SC55 Hurst, Fair Barbour (1-MS; University, MS); Reid, Talmage James (i-MS; University, MS) Ramsey numbers for cocircuits in matroids.  ...  Summary: “We determine upper bounds on the number of ele- ments in connected and 3-connected matroids with fixed rank and bounded cocircuit size.  ... 

Page 6037 of Mathematical Reviews Vol. , Issue 2000i [page]

2000 Mathematical Reviews  
The matroid Ramsey number n(a, db) is the least positive integer n such that every connected matroid with n elements has a circuit of size at least a or a cocircuit of size at least b.  ...  Sheng Jiang (PRC-YANG; Yangzhou) 2000i:05040 05B35 Bonin, Joseph E. (1-GWU; Washington, DC); McNulty, Jennifer (1-MT; Missoula, MT); Reid, Talmage James (1-MS; University, MS) The matroid Ramsey number  ... 

The Erdös–Pósa property for matroid circuits

Jim Geelen, Kasper Kabell
2009 Journal of combinatorial theory. Series B (Print)  
The number of disjoint cocircuits in a matroid is bounded by its rank.  ...  There are, however, matroids with arbitrarily large rank that do not contain two disjoint cocircuits; consider, for example, M (K n ) and U n,2n .  ...  Arranging circuits In this section we derive technical "Ramsey-like" results concerning arrangements of low rank sets in a matroid.  ... 
doi:10.1016/j.jctb.2008.08.004 fatcat:y6dqr77tqzanxhp2lhne5qjyc4

Fixing Numbers for Matroids

Gary Gordon, Jennifer McNulty, Nancy Ann Neudauer
2015 Graphs and Combinatorics  
Ramsey Number n(6,6) -Colloquium Talk, Washington State University, Pullman, WA, 1999 -Contributed Poster, Project NExT/Young Mathematician's Network Poster Session, AMS- MAA Joint Mathematics Meetings  ...  Neudauer, On cocircuit covers of bicircular matroids, Discrete Math, 308 (2008), no 17, 4008-4012. • F. Lutscher, J. McNulty, J. Morris, K.  ... 
doi:10.1007/s00373-015-1540-7 fatcat:7qbsws6rcvdn5k2y42l44wyxcq

A sharp bound on the size of a connected matroid

Manoel Lemos, James Oxley
2001 Transactions of the American Mathematical Society  
The second inequality is an interesting companion to Lehman's width-length inequality which asserts that the former inequality can be reversed for regular matroids when ce and c * e are replaced by the  ...  This paper proves that a connected matroid M whose largest circuit and largest cocircuit have c and c * elements, respectively, has at most 1 2 cc * elements.  ...  Acknowledgements The authors thank Guoli Ding for showing them a short proof of Theorem 1.5. The first author was partially supported by CNPq, CAPES, FINEP, and PRONEX 107/97.  ... 
doi:10.1090/s0002-9947-01-02767-2 fatcat:hxbblqhrcndzjfqg7undzzud3a

On a Generalization of Spikes

Nick Brettell, Rutger Campbell, Deborah Chun, Kevin Grace, Geoff Whittle
2019 SIAM Journal on Discrete Mathematics  
We show that for any positive integer t, there is a finite number of matroids with the (t, )-property for < 2t; however, matroids with the (t, 2t)-property form an infinite family.  ...  We consider matroids with the property that every subset of the ground set of size t is contained in both an -element circuit and an -element cocircuit; we say that such a matroid has the (t, )-property  ...  The authors would like to thank the Mathematical Research Institute (MATRIX), Creswick, Victoria, Australia, for support and hospitality during the Tutte Centenary Retreat, 26 Nov.-2 Dec. 2017, where work  ... 
doi:10.1137/18m1182255 fatcat:prkju5gkgfffzdn6sqi4mrxu2u

Page 3278 of Mathematical Reviews Vol. , Issue 85h [page]

1985 Mathematical Reviews  
Two different lower bounds are given for the total of the num- bers of pairwise disjoint circuits of M and of M*; a lower bound is also given for a number of pairwise disjoint 2-cocircuits of a minimally  ...  The author considers: (i) Hamiltonian circuits in graphs, (ii) finite Ramsey theory, (iii) Ulam’s conjecture. J.  ... 

On a generalisation of spikes [article]

Nick Brettell, Rutger Campbell, Deborah Chun, Kevin Grace, Geoff Whittle
2018 arXiv   pre-print
We show that for any positive integer t, there is a finite number of matroids with the (t,ℓ)-property for ℓ<2t; however, matroids with the (t,2t)-property form an infinite family.  ...  We consider matroids with the property that every subset of the ground set of size t is contained in both an ℓ-element circuit and an ℓ-element cocircuit; we say that such a matroid has the (t,ℓ)-property  ...  We define the function h k : N → N, for each k ∈ [t], as follows: h k (t) = 4t − 3 if k = t, r k (h k+1 (t)) if k ∈ [t − 1], where r k (n) is the Ramsey number described in Theorem 5.7.  ... 
arXiv:1804.06959v1 fatcat:3q4burnoofdg5c2dltdeca6tju

Page 2012 of Mathematical Reviews Vol. , Issue 96d [page]

1996 Mathematical Reviews  
Reid (1-MS; University, MS) 96d:05033 05B35 0SD10 Hurst, Fair Barbour (1-MS; University, MS); Reid, Talmage James (1-MS; University, MS) Some small circuit—cocircuit Ramsey numbers for matroids.  ...  An extremal matroid of a number n(k,/) is a connected matroid M with n(k,/)—1 elements such that each circuit of M has fewer than k elements and each cocircuit of M has fewer than / elements.  ... 

Page 6355 of Mathematical Reviews Vol. , Issue 2002I [page]

2002 Mathematical Reviews  
We also give a lower bound on the number of elements meeting some 2-element cocircuit in terms of the total number of elements in a minimally 2-connected matroid.  ...  Recent results on connectivity, cycle covers, removable circuits, graph minors, branch-width, representability, and Ramsey theory for graphs and matroids are presented here.  ... 

Page 4779 of Mathematical Reviews Vol. , Issue 98H [page]

1998 Mathematical Reviews  
The matroid Ramsey number n(k,/) is the least positive integer n such that every connected matroid on n elements contains either a circuit with at least k elements or a cocircuit with at least / elements  ...  The purpose of the paper is to illustrate the similarities between the Ramsey numbers and the matroid Ramsey numbers.  ... 

Some heterochromatic theorems for matroids [article]

Criel Merino, Juan José Montellano-Ballesteros
2017 arXiv   pre-print
We also extend the trivial observation of Erdös, Simonovits and Sós for the anti-Ramsey number for 3-cycles to 3-circuits in projective geometries over finite fields.  ...  The anti-Ramsey number of Erdös, Simonovits and Sós from 1973 has become a classic invariant in Graph Theory.  ...  this work we try to explore the natural generalization of the anti-Ramsey number for graphs to matroids, using the known generalization of heterochromatic number in hypergraphs.  ... 
arXiv:1708.08562v1 fatcat:sakk24q2ynalhhxp33yaym2egm

Page 7209 of Mathematical Reviews Vol. , Issue 96m [page]

1996 Mathematical Reviews  
The authors prove a Ramsey theorem for binary matroids.  ...  Interesting Ramsey results on matrices are developed and used in the proof of the above theorem. This result generalizes a similar theorem for graphs of Oporowski, Oxley, and R. Thomas [J. Combin.  ... 

Cyclic Polytopes and Oriented Matroids

Raul Cordovil, Pierre Duchet
2000 European journal of combinatorics (Print)  
., those linear orderings of the vertices for which Gale's evenness criterion holds. Proofs give a systematic account on an oriented matroid approach to cyclic polytopes.  ...  Consider the moment curve in the real euclidean space R d defined parametrically by the map is the convex hull of n > d different points on this curve.  ...  Ziegler for inviting us to participate in the present volume.  ... 
doi:10.1006/eujc.1999.0317 fatcat:jhixy5qjpfeohpfxai7hjlllyq
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