Rota's Conjecture Research Articles

A generalisation of uniform matroids

A matroid is uniform if and only if it has no minor isomorphic to U1,1⊕U0,1 and is paving if and only if it has no minor isomorphic to U2,2⊕U0,1. This paper considers, more generally, when a matroid M has no Uk,k⊕U0,ℓ-minor for a fixed pair of positive integers (k,ℓ). Calling such a matroid (k,ℓ)-uniform, it is shown that this is equivalent to the condition that every rank-(r(M)−k) flat of M has nullity less than ℓ. Generalising a result of Rajpal, we prove that for any pair (k,ℓ) of positive integers and prime power q, only finitely many simple cosimple GF(q)-representable matroids are (k,ℓ)-uniform. Consequently, if Rota's Conjecture holds, then for every prime power q, there exists a pair (kq,ℓq) of positive integers such that every excluded minor of GF(q)-representability is (kq,ℓq)-uniform. We also determine all binary (2,2)-uniform matroids and show the maximally 3-connected members to be Z5﹨t,AG(4,2),AG(4,2)⁎ and a particular self-dual matroid P10. Combined with results of Acketa and Rajpal, this completes the list of binary (k,ℓ)-uniform matroids for which k+ℓ≤4.

Matroid fragility and relaxations of circuit hyperplanes

Abstract We relate two conjectures that play a central role in the reported proof of Rota's Conjecture. Let F be a finite field. The first conjecture states that: the branch-width of any F -representable N-fragile matroid is bounded by a function depending only upon F and N. The second conjecture states that: if a matroid M 2 is obtained from a matroid M 1 by relaxing a circuit-hyperplane and both M 1 and M 2 are F -representable, then the branch-width of M 1 is bounded by a function depending only upon F . Our main result is that the second conjecture implies the first.

On Rota's Conjecture and nested separations in matroids

We prove that for each finite field F and integer k∈Z there exists n∈Z such that no excluded minor for the class of F-representable matroids has n nested k-separations.

The odd case of Rota's bases conjecture

The paper links four conjectures:(1)(Rota's bases conjecture): For any system A=(A1,…,An) of non-singular real valued matrices the multiset of all columns of matrices in A can be decomposed into n independent systems of representatives of A.(2)(Alon–Tarsi): For even n, the number of even n×n Latin squares differs from the number of odd n×n Latin squares.(3)(Stones–Wanless, Kotlar): For all n, the number of even n×n Latin squares with the identity permutation as first row and first column differs from the number of odd n×n Latin squares of this type.(4)(Aharoni–Berger): Let M and N be two matroids on the same vertex set, and let A1,…,An be sets of size n+1 belonging to M∩N. Then there exists a set belonging to M∩N meeting all Ai. Huang and Rota [8] and independently Onn [11] proved that for any n (2) implies (1). We prove equivalence between (2) and (3). Using this, and a special case of (4), we prove the Huang–Rota–Onn theorem for n odd and a restricted class of input matrices: assuming the Alon–Tarsi conjecture for n−1, Rota's conjecture is true for any system of non-singular real valued matrices where one of them is non-negative and the remaining have non-negative inverses.

IS THE MISSING AXIOM OF MATROID THEORY LOST FOREVER?

We conjecture that it is not possible to finitely axiomatize matroid representability in monadic second-order logic (MSOL) for matroids, and we describe some partial progress towards this conjecture. We present a collection of sentences in MSOL and show that it is possible to finitely axiomatize matroids using only sentences in this collection. Moreover, we can also axiomatize representability over any fixed finite field (assuming Rota's conjecture holds). We prove that it is not possible to finitely axiomatize representability, or representability over any fixed infinite field, using sentences from the collection.

Solving Rota's Conjecture

I n 1970, Gian-Carlo Rota posed a conjecture predicting a beautiful combinatorial characterization of linear dependence in vector spaces over any given finite field. We have recently completed a fifteen-year research program that culminated in a solution to Rota’s Conjecture. In this article we discuss the conjecture and give an overview of the proof. Matroids are a combinatorial abstraction of linear independence among vectors; given a finite collection of vectors in a vector space, each subset is either dependent or independent. A matroid consists of a finite ground set together with a collection of subsets that we call independent; the independent sets satisfy natural combinatorial axioms coming from linear algebra. Not all matroids can be represented by a collection of vectors and, ever since their introduction by Hassler Whitney [26] in 1935, mathematicians have sought ways to characterize those matroids that are. Rota’s Conjecture asserts that representability over any given finite field is characterized by a finite list of obstructions. We will formalize these notions, and the conjecture, in the next section. In the remainder of this introduction, we will describe the journey that led us to a solution. In the late 1990s, Rota’s Conjecture was already known to hold for fields of size two, three, and

On Rota's conjecture and excluded minors containing large projective geometries

We prove that an excluded minor for the class of GF ( q ) -representable matroids cannot contain a large projective geometry over GF ( q ) as a minor.

Fork-decompositions of matroids

One of the central problems in matroid theory is Rota's conjecture that, for all prime powers q, the class of GF( q)-representable matroids has a finite set of excluded minors. This conjecture has been settled for q⩽4 but remains open otherwise. Further progress towards this conjecture has been hindered by the fact that, for all q>5, there are 3-connected GF( q)-representable matroids having arbitrarily many inequivalent GF( q)-representations. This fact refutes a 1988 conjecture of Kahn that 3-connectivity would be strong enough to ensure an absolute bound on the number of such inequivalent representations. This paper introduces fork-connectivity, a new type of self-dual 4-connectivity, which we conjecture is strong enough to guarantee the existence of such a bound but weak enough to allow for an analogue of Seymour's Splitter Theorem. We prove that every fork-connected matroid can be reduced to a vertically 4-connected matroid by a sequence of operations that generalize Δ– Y and Y– Δ exchanges. It follows from this that the analogue of Kahn's Conjecture holds for fork-connected matroids if and only if it holds for vertically 4-connected matroids. The class of fork-connected matroids includes the class of 3-connected forked matroids. By taking direct sums and 2-sums of matroids in the latter class, we get the class M of forked matroids, which is closed under duality and minors. The class M is a natural subclass of the class of matroids of branch-width at most 3 and includes the matroids of path-width at most 3. We give a constructive characterization of the members of M and prove that M has finitely many excluded minors.

Branch-Width and Rota's Conjecture

For a fixed finite field F and an integer k there are a finite number of matroids of branch-width k that are excluded minors for F -representability.

An exchange property of matroid

Rota's conjecture about n bases in a rank n matroid is solved for n = 3.

On the Dinitz conjecture and related conjectures

We present previously unpublished elementary proofs by Dekker and Ottens (1991) and Boyce (private communication) of a special case of the Dinitz conjecture. We prove a special case of a related basis conjecture by Rota, and give a reformulation of Rota's conjecture using the Nullstellensatz. Finally we give an asymptotic result on a related Latin square conjecture.

An infinitary version of the Graham-Leeb-Rothschild theorem

This paper contains a proof of a conjecture of S. Simpson which is an infinitary version of a conjecture of Rota concerning partitions of finite dimensional vector spaces over a finite field. Rota's conjecture was proved by Graham, Leeb, and Rothschild ( Adv. in Math. 8 (1972), 417–433).

A Generalized Rota Conjecture for Partitions

A Generalized Rota Conjecture for Partitions

Ramsey's Theorem for a Class of Categories

Ramsey's Theorem states that for a sufficiently large set S, and for any splitting of the k-element subsets of S into r classes, there is a subset T [unk] S, [unk]T[unk] = l, such that all k-element subsets of T are in the same class. This paper establishes a theorem for certain categories that generalizes Ramsey's Theorem. In particular, it is strong enough to establish G-C. Rota's conjecture that the vector space analogue to Ramsey's Theorem is true. It also implies the Ramsey theorem for n-parameter sets, which has as corollaries, among others, the theorem of van der Waerden on arithmetic progressions and several results of R. Rado on regularity in systems of linear equations.