I received the sad news that Dominic Welsh passed away last week. Here is Peter Cameron’s blog post.

https://cameroncounts.wordpress.com/2023/12/04/dominic-welsh/

]]>Thanks! I’ve added a link to Combinatorial Theory in the post.

]]>For a line L in rank-3 matroid M on ground-set E, we make a symmetric ExE matrix A(L) where the (i,j)-entry is 1 if i and j are in L, and 0 otherwise. Adding the matrices A(L) gives a matrix a symmetric ExE matrix whose off-diagonal entries are all 1, and whose diagonal entries are at least 2. It is easy to see that all such matrices are nonsingular. However each of the matrices A(L) have rank 1. So the number of lines is at least |E|.

]]>In some sense, a multimatroid already contains all of its duals. More precisely, if you construct a 2-matroid (or a 3-matroid) from a delta-matroid D, then the multimatroid M encodes D and all of its twists (twisted duals). Really, M depends on the collection of twists of D rather than D itself. To get D back from M, you have to remember what E_1 and E_2 are. Varying the partition of the elements into E_1 and E_2, and reversing the construction gives you all twists of D.

]]>The definition of a convex hull seems to only include the line segments between the points in S. So for example, with the current definition the covex hull of three (non colinear) points would be the lines of the triangle that they define, but would not include the interior. Am I missing something?

]]>On a related note, is there a way to generalise duality to multimatroids/p-matroids? Some desiderata of such a function would be that it’s an involution between multimatroids and that minors of duals are equal to duals of minors.

]]>Curtis Greene (A rank inequality for finite geometric lattices, https://www.sciencedirect.com/science/article/pii/S0021980070800904) showed that a simple matroid has the same number of points as hyperplanes if and only if it is modular.

Modular matroids are known to be direct sums of projective geometries and a free matroid (see e.g. Proposition 6.9.1 in James’ book). Hence, for rank at least 4, the only simple matroid for which the numbers of points and hyperplanes are equal is the free matroid $U_{n,n}$.

So the answer to my question is “no, unless the matroid is a free matroid”.

Thanks, James!

]]>I am looking for a general context in Matroid theory in which any finite rank closed

set cannot be written as a finite union of closed sets of of smaller rank.

Thank you. ]]>

False conjecture: Every simple rank-4 matroid with no lines of length 4 or more contains a plane with at most 4 points.

By results of Keevash on the existence of designs, for each k and infinitely many n there exists a Steiner systems S(3,k,n); this is a collection of k-element subsets of an n-element set such that each triple is contained in exactly one of the subsets in the collection. These k-element subsets define the planes of an n-element rank-4 paving matroid. That matroid is triagnle-free and all of its planes have exactly k points.

]]>Thanks Michael. How confident are you about this? Sorry about my very slow reply, I only just saw your response.

Cheers,

James

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