Related papers: Two new results on maximal left-compressed interse…

This manuscript explores many convolution (restricted summation) type sequences via certain types of matrix based factorizations that can be used to express their generating functions. The last primary (non-appendix) section of the thesis…

Three intersection theorems are proved. First, we determine the size of the largest set system, where the system of the pairwise unions is l-intersecting. Then we investigate set systems where the union of any s sets intersect the union of…

In 1984, Frankl and Pach proved that, for positive integers $n$ and $d$, the maximum size of a $(d+1)$-uniform set family $\mathcal{F}$ on an $n$-element set with VC-dimension at most $d$ is at most ${n\choose d}$; and they suspected that…

For $n > 2k \geq 4$ we consider intersecting families $\mathcal F$ consisting of $k$-subsets of $\{1, 2, \ldots, n\}$. Let $\mathcal I(\mathcal F)$ denote the family of all distinct intersections $F \cap F'$, $F \neq F'$ and $F, F'\in…

We establish a connection between the problem of constructing maximal collections of mutually unbiased bases (MUBs) and an open problem in the theory of Lie algebras. More precisely, we show that a collection of m MUBs in K^n gives rise to…

In this paper, we address several Erd\H os--Ko--Rado type questions for families of partitions. Two partitions of $[n]$ are {\it $t$-intersecting} if they share at least $t$ parts, and are {\it partially $t$-intersecting} if some of their…

We propose a unified theory of generalized weights for linear codes endowed with an arbitrary distance. Instead of relying on supports or anticodes, the weights of a code are defined via the intersections of the code with a chosen family of…

The classical Erd\H os-Ko-Rado (EKR) Theorem states that if we choose a family of subsets, each of size (k), from a fixed set of size (n (n > 2k)), then the largest possible pairwise intersecting family has size (t ={n-1\choose k-1}). We…

We determine the size of spaces of higher order Maass forms of even weight for cofinite discrete subgroups of PSL(2,R) with cusps. If exponential growth at the cusps is allowed, the spaces of Maass forms of a given order are as large as…

This paper proves a stability result for a variation of the Erd\H{o}s-Ko-Rado theorem in the context of polynomials over finite fields. Let $\mathcal F$ be a family of polynomials of degree at most $k \geq 3$ in $\mathbb F_q[X]$. Call…

In this paper, we address several intersection problems for $r$-cross $t$-intersecting families of partitions. A $k$-partition of an $n$-set $X$ is a set of $k$ pairwise disjoint non-empty subsets whose union is $X$. For $1\leq i\leq r$,…

We relate the maximum semidefinite and linear extension complexity of a family of polytopes to the cardinality of this family and the minimum pairwise Hausdorff distance of its members. This result directly implies a known lower bound on…

We introduce $k$-maximal sub-polygon graphs ($k$-MSP graphs), the intersection graphs of maximal polygons contained in a polygon with sides parallel to a regular $2k$-gon. We prove that all complete graphs are $k$-MSP graphs for all $k>1$;…

Every minor-closed class of matroids of bounded branch-width can be characterized by a list of excluded minors, but unlike graphs, this list may need to be infinite in general. However, for each fixed finite field $\mathbb F$, the list…

An $(n, k_1, \dots, k_t)$-cross intersecting system is a set of non-empty pairwise cross-intersecting families $\mathcal{F}_1\subset{[n]\choose k_1}, \mathcal{F}_2\subset{[n]\choose k_2}, \dots, \mathcal{F}_t\subset{[n]\choose k_t}$ with…

We investigate the combinatorial structure of subspaces of the exterior algebra of a finite-dimensional real vector space, working in parallel with the extremal combinatorics of hypergraphs. Using initial monomials, projections of the…

Motivated by odd-sunflowers, introduced recently by Frankl, Pach, and P{\'a}lv{\"o}lgyi, we initiate the study of temperate families: a family $\mathcal{F} \subseteq \mathcal{P}([n])$ is said to be \emph{temperate} if each $A \in…

We study augmentations of a Legendrian surface $L$ in the $1$-jet space, $J^1M$, of a surface $M$. We introduce two types of algebraic/combinatorial structures related to the front projection of $L$ that we call chain homotopy diagrams…

A flag of a finite set $S$ is a set $f$ of non-empty, proper subsets of $S$, such that $X\subseteq Y$ or $Y\subseteq X$ for all $X,Y\in f$. Two flags $f_1$ and $f_2$ of $S$ are opposite if $X_1\cap X_2=\emptyset$, or $X_1\cup X_2=S$ for all…

A Fourier restriction estimate is obtained for a broad class of conic surfaces by adding a weight to the usual underlying measure. The new restriction estimate exhibits a certain affine-invariance and implies the sharp $L^p-L^q$ restriction…