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References (24)
-
S. Jukna (2011)
Extremal combinatorics: with applications in computer science -
D. Knuth (1974)
The Asymptotic Number of GeometriesJ. Comb. Theory, Ser. A, 16
-
W. Samotij (2014)
Counting independent sets in graphsEur. J. Comb., 48
-
David Ellis (2019)
Extremal Combinatorics -
D. Kleitman, Kenneth Winston (1982)
On the number of graphs without 4-cyclesDiscret. Math., 41
-
A. E. Brouwer, A.M. Cohen, A. Neumaier (1989)
Distance-Regular Graphs -
William Critchlow (2016)
Minors of asymptotically almost all sparse paving matroidsarXiv: Combinatorics
-
P. Nelson (2016)
Almost all matroids are nonrepresentableBulletin of the London Mathematical Society, 50
-
N. Bansal, R. Pendavingh, J. Pol (2012)
On the number of matroidsCombinatorica, 35
-
Dillon Mayhew, Mike Newman, D. Welsh, G. Whittle (2011)
On the asymptotic proportion of connected matroidsEur. J. Comb., 32
-
(2009)
On excluded minors for realrepresentability -
J. Geelen, B. Gerards, G. Whittle (2014)
Solving Rota's ConjectureNotices of the American Mathematical Society, 61
-
Ihrer Grenzgebiete, Theorie Der, Konvexen Körper (1975)
Ergebnisse der Mathematik und ihrer GrenzgebieteSums of Independent Random Variables
-
N. Biggs (1972)
ON THE FOUNDATIONS OF COMBINATORIAL THEORY: COMBINATORIAL GEOMETRIESBulletin of The London Mathematical Society, 4
-
L. Lowrance, J. Oxley, C. Semple, D. Welsh (2013)
On properties of almost all matroidsAdv. Appl. Math., 50
-
B. Cloteaux (2010)
Approximating the number of bases for almost all matroidsCongressus Numerantium, 202
-
J. Oxley (2011)
Matroid theory -
S. Jukna (2003)
Extremal Combinatorics: with Applications in Computer Science by Stasys Jukna, Springer, 2001, xvii + 375 pp. 32.50; $49.95, ISBN 3540663134 -
F. Graham, R. Graham, P. Frankl, J. Shearer (1986)
Some intersection theorems for ordered sets and graphsJ. Comb. Theory, Ser. A, 43
-
Pendavingh, van Pol (2017)
On the number of matroids compared to the number of sparse paving matroids -
R. Pendavingh, J. Pol (2013)
Counting matroids in minor-closed classesJ. Comb. Theory, Ser. B, 111
-
Brian Cloteaux (2011)
Approximating the Number of Bases for Almost All MatroidsInformation Processing Letters
-
(2011)
Matroid theory, volume 21 of Oxford Graduate Texts in Mathematics -
R. Graham, N. Sloane (1980)
Lower bounds for constant weight codesIEEE Trans. Inf. Theory, 26
- Publisher
- Springer Journals
- Copyright
- Copyright © 2018 by János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature
- Subject
- Mathematics; Combinatorics; Mathematics, general
- ISSN
- 0209-9683
- eISSN
- 1439-6912
- DOI
- 10.1007/s00493-016-3594-4
- Publisher site
- See Article on Publisher Site
Abstract
For a matroid M of rank r on n elements, let b(M) denote the fraction of bases of M among the subsets of the ground set with cardinality r. We show that $$\Omega \left( {1/n} \right) \leqslant 1 - b\left( M \right) \leqslant O\left( {\log {{\left( n \right)}^3}/n} \right)a\;sn \to \infty $$ Ω ( 1 / n ) ≤ 1 − b ( M ) ≤ O ( log ( n ) 3 / n ) a s n → ∞ for asymptotically almost all matroids M on n elements. We derive that asymptotically almost all matroids on n elements (1) have a U k,2k -minor, whenever k≤O(log(n)), (2) have girth ≥Ω(log(n)), (3) have Tutte connectivity $$\geq\Omega\;{(\sqrt {log(n)})}$$ ≥ Ω ( l o g ( n ) ) , and (4) do not arise as the truncation of another matroid.
Journal
Combinatorica – Springer Journals
Published: Jun 5, 2018