An elementary, illustrative proof of the Rado-Horn Theorem

The Rado-Horn theorem provides necessary and sufficient conditions for when a collection of vectors can be partitioned into a fixed number of linearly independent sets. Such partitions exist if and only if every subset of the vectors satisfies the so-called Rado-Horn inequality. Today there are at least six proofs of the Rado-Horn theorem, but these tend to be extremely delicate or require intimate knowledge of matroid theory. In this paper we provide an elementary proof of the Rado-Horn theorem as well as elementary proofs for several generalizations including results for the redundant case when the hypotheses of the Rado-Horn theorem fail. Another problem with the existing proofs of the Rado-Horn Theorem is that they give no information about how to actually partition the vectors. We start by considering a specific partition of the vectors, and the proof consists of showing that this is an optimal partition. We further show how certain structures we construct in the proof are at the heart of the Rado-Horn theorem by characterizing subsets of vectors which maximize the Rado-Horn inequality. Lastly, we demonsrate how these results may be used to select an optimal partition with respect to spanning properties of the vectors

Intersecting families of discrete structures are typically trivial

The study of intersecting structures is central to extremal combinatorics. A family of permutations $\mathcal{F} \subset S_n$ is \emph{$t$-intersecting} if any two permutations in $\mathcal{F}$ agree on some $t$ indices, and is \emph{trivial} if all permutations in $\mathcal{F}$ agree on the same $t$ indices. A $k$-uniform hypergraph is \emph{$t$-intersecting} if any two of its edges have $t$ vertices in common, and \emph{trivial} if all its edges share the same $t$ vertices. The fundamental problem is to determine how large an intersecting family can be. Ellis, Friedgut and Pilpel proved that for $n$ sufficiently large with respect to $t$, the largest $t$-intersecting families in $S_n$ are the trivial ones. The classic Erd\H{o}s--Ko--Rado theorem shows that the largest $t$-intersecting $k$-uniform hypergraphs are also trivial when $n$ is large. We determine the \emph{typical} structure of $t$-intersecting families, extending these results to show that almost all intersecting families are trivial. We also obtain sparse analogues of these extremal results, showing that they hold in random settings. Our proofs use the Bollob\'as set-pairs inequality to bound the number of maximal intersecting families, which can then be combined with known stability theorems. We also obtain similar results for vector spaces.Comment: 19 pages. Update 1: better citation of the Gauy--H\`an--Oliveira result. Update 2: corrected statement of the unpublished Hamm--Kahn result, and slightly modified notation in Theorem 1.6 Update 3: new title, updated citations, and some minor correction

Conditions for matchability in groups and vector spaces II

We present sufficient conditions for the existence of matchings in abelian groups and their linear counterparts. These conditions lead to extensions of existing results in matching theory. Additionally, we classify subsets within abelian groups that cannot be matched. We introduce the concept of Chowla subspaces and formulate and conjecture a linear analogue of a result originally attributed to Y. O. Hamidoune [20] concerning Chowla sets. If proven true, this result would extend matchings in primitive subspaces. Throughout the paper, we emphasize the analogy between matchings in abelian groups and field extensions. We also pose numerous open questions for future research. Our approach relies on classical theorems in group theory, additive number theory and linear algebra. As the title of the paper suggests, this work is the second sequel to a previous paper [5] with a similar theme. This paper is self-contained and can be read independently.Comment: Comments are welcom

Enumerating matroid extensions

This thesis investigates the problem of enumerating the extensions of certain matroids. A matroid M is an extension of a matroid N if M delete e is equal to N for some element e of M. Similarly, a matroid M is a coextension of a matroid N if M contract e is equal to N for some element e of M. In this thesis, we consider extensions and coextensions of matroids in the classes of graphic matroids, representable matroids, and frame matroids. We develop a general strategy for counting the extensions of matroids which translates the problem into counting stable sets in an auxiliary graph. We apply this strategy to obtain asymptotic results on the number of extensions and coextensions of certain graphic matroids, projective geometries, and Dowling geometries

Discrete Geometry

The workshop on Discrete Geometry was attended by 53 participants, many of them young researchers. In 13 survey talks an overview of recent developments in Discrete Geometry was given. These talks were supplemented by 16 shorter talks in the afternoon, an open problem session and two special sessions. Mathematics Subject Classification (2000): 52Cxx. Abstract regular polytopes: recent developments. (Peter McMullen) Counting crossing-free configurations in the plane. (Micha Sharir) Geometry in additive combinatorics. (József Solymosi) Rigid components: geometric problems, combinatorial solutions. (Ileana Streinu) • Forbidden patterns. (János Pach) • Projected polytopes, Gale diagrams, and polyhedral surfaces. (Günter M. Ziegler) • What is known about unit cubes? (Chuanming Zong) There were 16 shorter talks in the afternoon, an open problem session chaired by Jesús De Loera, and two special sessions: on geometric transversal theory (organized by Eli Goodman) and on a new release of the geometric software Cinderella (Jürgen Richter-Gebert). On the one hand, the contributions witnessed the progress the field provided in recent years, on the other hand, they also showed how many basic (and seemingly simple) questions are still far from being resolved. The program left enough time to use the stimulating atmosphere of the Oberwolfach facilities for fruitful interaction between the participants