On the number of matroids

Bansal, Nikhil; Pendavingh, Rudi; Pol, J.G. van der

doi:10.1007/s00493-014-3029-z

Cited by 29 publications

(73 citation statements)

References 29 publications

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“…In Section 4, we confirm this conjecture for the following sparse paving matroids N : the uniform matroids U 2,k and U 3,6 , and the matroids P 6 , Q 6 , and R 6 (see Fig. 1).…”

Section: Introductionsupporting

confidence: 65%

“…As a corollary, we obtain the weaker of the two upper bounds on the number of matroids that appeared in [3]. Corollary 1. log log m n ≤ n − (3/2) log n + 2 log log n + O(1).…”

Section: Theorem 4 Let M Be a Contraction-closed Class Of Matroids mentioning

confidence: 85%

“…Suppose M ∈ M n,3 and M U 3,6 . If M := si(M ) has at most 55 elements, then M has at most If | | = 4, and there is some f ∈ E(M ) \ ( ∪ ) that is on at most one line which intersects both \ {e} and \ {e}, then we may choose {e, f, p, q, p , q } spanning a Q 6 -minor as before.…”

Section: Lemma 14 Let M Be a Simple Matroid Of Rank 3 With A U 3k -mentioning

confidence: 99%

“…Where it is straightforward to construct a random graph, it is not at all obvious what a good model of random matroids should look like. In [3], Nikhil Bansal and the present authors proved an upper bound on the number m n of distinct matroids on n elements, the denominator in (1). Combining our upper bound with the lower bound due to Knuth [10], we have 1 n n n/2 ≤ log m n ≤ 2 n n n/2 1 + o(1) as n → ∞.…”

Section: Introductionmentioning

confidence: 99%

“…We hope that by tightening the gap between the upper and lower bound in (2), we will eventually arrive at a satisfactory model of random matroids as well. Meanwhile, the methods of [3] do inspire the following general strategy for showing that asymptotically almost every matroid has a certain property P. The argument for bounding the size of a concise description of a matroid M can sometimes be extended to show that if M does not have a property P, then M has a description of a much shorter length than the general upper bound in (2). Then, the upper bound on the number of matroids without property P is much better accordingly.…”

Section: Introductionmentioning

confidence: 99%

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Counting matroids in minor-closed classes

Pendavingh

Pol

2015

Journal of Combinatorial Theory, Series B

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A flat cover is a collection of flats identifying the non-bases of a matroid. We introduce the notion of cover complexity, the minimal size of such a flat cover, as a measure for the complexity of a matroid, and present bounds on the number of matroids on n elements whose cover complexity is bounded. We apply cover complexity to show that the class of matroids without an N -minor is asymptotically small in case N is one of the sparse paving matroids U 2,k , U 3,6 , P 6 , Q 6 or R 6 , thus confirming a few special cases of a conjecture due to Mayhew, Newman, Welsh, and Whittle. On the other hand, we show a lower bound on the number of matroids without an M (K 4 )-minor which asymptotically matches the best known lower bound on the number of all matroids, due to Knuth.

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“…In Section 4, we confirm this conjecture for the following sparse paving matroids N : the uniform matroids U 2,k and U 3,6 , and the matroids P 6 , Q 6 , and R 6 (see Fig. 1).…”

Section: Introductionsupporting

confidence: 65%

“…As a corollary, we obtain the weaker of the two upper bounds on the number of matroids that appeared in [3]. Corollary 1. log log m n ≤ n − (3/2) log n + 2 log log n + O(1).…”

Section: Theorem 4 Let M Be a Contraction-closed Class Of Matroids mentioning

confidence: 85%

Section: Lemma 14 Let M Be a Simple Matroid Of Rank 3 With A U 3k -mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Counting matroids in minor-closed classes

Pendavingh

Pol

2015

Journal of Combinatorial Theory, Series B

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Minors of a random binary matroid

Cooper

Frieze

Pegden

2019

Random Struct Algorithms

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Let A be an n × m matrix over GF 2 where each column consists of k ones, and let M be an arbitrary fixed binary matroid. The matroid growth rate theorem implies that there is a constant C M such that m ≥ C M n 2 implies that the binary matroid induced by A contains M as a minor. We prove that if the columns of A = A n,m,k are chosen randomly, then there are constants k M , L M such that k ≥ k M and m ≥ L M n implies that A contains M as a minor with high probability.KEYWORDS binary, matroids, minors, random 1 Random Struct Alg. 2019;55:865-880.wileyonlinelibrary.com/journal/rsa

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On the Number of Bases of Almost All Matroids

Pendavingh

Pol

2018

Combinatorica

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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 thatfor 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 ≥ Ω( log(n)), and (4) do not arise as the truncation of another matroid.Our argument is based on a refined method for writing compressed descriptions of any given matroid, which allows bounding the number of matroids in a class relative to the number of sparse paving matroids.Conjecture 1.1. As n → ∞, asymptotically almost all matroids on n elements are sparse paving.If this conjecture were true, then several other asymptotic properties of matroids would follow with little extra work, as it would suffice to establish the property for almost all sparse paving matroids. For example, almost all sparse paving matroids contain a fixed uniform matroid, are highly connected, have high girth, and the analogous statements for general matroids would follow. We next describe our intuition supporting Conjecture 1.1, which we partially substantiated to obtain the results of this paper.Consider the Johnson graph J(E, r), whose vertices are the subsets of E of cardinality r, and in which two vertices X, Y are adjacent exactly if |X ∩ Y | = r − 1. The Johnson graph will serve as an 'ambient space' for all the matroids on ground set E and of rank r. In what follows, we will write G := J(E, r).

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On the number of matroids

Cited by 29 publications

References 29 publications

Counting matroids in minor-closed classes

Counting matroids in minor-closed classes

Minors of a random binary matroid

On the Number of Bases of Almost All Matroids

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