The Maximum Size of Combinatorial Geometries Excluding Wheels and Whirls as Minors

We show that the maximum size of a geometry of rank n excluding the (q + 2)-point line, the 3-wheel W_3, and the 3-whirl W^3 as minor is (n - 1)q + 1, and geometries of maximum size are parallel connections of (q + 1)-point lines. We show that the maximum size of a geometry of rank n excluding the 5-point line, the 4-wheel W_4, and the 4-whirl W^4 as minors is 6n - 5, for n ≥ 3. Examples of geometries having rank n and size 6n - 5 include parallel connections of the geometries V_19 and PG(2,3)

Upper bounds on Ramsey numbers for vector spaces over finite fields

For $B \subseteq \mathbb F_q^m$, let $\mathrm{ex}_{\mathrm{aff}}(n,B)$ denote the maximum cardinality of a set $A \subseteq \mathbb F_q^n$ with no subset which is affinely isomorphic to $B$. Furstenberg and Katznelson proved that for any $B \subseteq \mathbb F_q^m$, $\mathrm{ex}_{\mathrm{aff}}(n,B)=o(q^n)$ as $n \to \infty$. For certain $q$ and $B$, some more precise bounds are known. We connect some of these problems to certain Ramsey-type problems, and obtain some new bounds for the latter. For $s,t \geq 1$, let $R_q(s,t)$ denote the minimum $n$ such that in every red-blue coloring of one-dimensional subspaces of $\mathbb F_q^n$, there is either a red $s$-dimensional subspace of $\mathbb F_q^n$ or a blue $t$-dimensional subspace of $\mathbb F_q^n$. The existence of these numbers is implied by the celebrated theorem of Graham, Leeb, Rothschild. We improve the best known upper bounds on $R_2(2,t)$, $R_3(2,t)$, $R_2(t,t)$, and $R_3(t,t)$

T-uniqueness of some families of k-chordal matroids

We define k-chordal matroids as a generalization of chordal matroids, and develop tools for proving that some k-chordal matroids are T-unique, that is, determined up to isomorphism by their Tutte polynomials. We apply this theory to wheels, whirls, free spikes, binary spikes, and certain generalizations.Postprint (published version

Finding a perfect matching of F2n\mathbb{F}_2^n with prescribed differences

We consider the following question by Balister, Gy\H{o}ri and Schelp: given $2^{n-1}$ nonzero vectors in $\mathbb{F}_2^n$ with zero sum, is it always possible to partition the elements of $\mathbb{F}_2^n$ into pairs such that the difference between the two elements of the $i$-th pair is equal to the $i$-th given vector for every $i$? An analogous question in $\mathbb{F}_p$, which is a case of the so-called "seating couples" problem, has been resolved by Preissmann and Mischler in 2009. In this paper, we prove the conjecture in $\mathbb{F}_2^n$ in the case when the number of distinct values among the given difference vectors is at most $n-2\log n-1$, and also in the case when at least a fraction $\frac12+\varepsilon$ of the given vectors are equal (for all $\varepsilon>0$ and $n$ sufficiently large based on $\varepsilon$).Comment: 18 page

Induced Binary Submatroids

The notion of induced subgraphs is extensively studied in graph theory. An example is the famous Gy\'{a}rf\'{a}s-Sumner conjecture, which asserts that given a tree $T$ and a clique $K$, there exists a constant $c$ such that the graphs that omit both $T$ and $K$ as induced subgraphs have chromatic number at most $c$. This thesis aims to prove natural matroidal analogues of such graph-theoretic problems