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Ramsey Numbers for Matroids

1997
*
European journal of combinatorics (Print)
*

It is natural then to construct

doi:10.1006/eujc.1995.0112
fatcat:ojb25uzevvgstnxglh5xpq4amq
*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*. ...##
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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*...##
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The Erdös–Pósa property for matroid circuits

2009
*
Journal of combinatorial theory. Series B (Print)
*

The

doi:10.1016/j.jctb.2008.08.004
fatcat:y6dqr77tqzanxhp2lhne5qjyc4
*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*. ...##
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Fixing Numbers for Matroids

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. ...

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A sharp bound on the size of a connected matroid

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

doi:10.1090/s0002-9947-01-02767-2
fatcat:hxbblqhrcndzjfqg7undzzud3a
*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. ...##
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On a Generalization of Spikes

2019
*
SIAM Journal on Discrete Mathematics
*

We show that

doi:10.1137/18m1182255
fatcat:prkju5gkgfffzdn6sqi4mrxu2u
*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 ...##
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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. ...##
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On a generalisation of spikes
[article]

2018
*
arXiv
*
pre-print

We show that

arXiv:1804.06959v1
fatcat:3q4burnoofdg5c2dltdeca6tju
*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. ...##
###
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*. ...##
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Some heterochromatic theorems for matroids
[article]

2017
*
arXiv
*
pre-print

We also extend the trivial observation of Erdös, Simonovits and Sós

arXiv:1708.08562v1
fatcat:sakk24q2ynalhhxp33yaym2egm
*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. ...##
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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. ...##
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Cyclic Polytopes and Oriented Matroids

2000
*
European journal of combinatorics (Print)
*

., those linear orderings of the vertices

doi:10.1006/eujc.1999.0317
fatcat:jhixy5qjpfeohpfxai7hjlllyq
*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. ...
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