Related papers: A few new oddtown and eventown problems

Two families $\mathcal{F},\mathcal{G}$ of $k$-subsets of $\{1,2,\ldots,n\}$ are called non-trivial cross-intersecting if $F\cap G\neq \emptyset$ for all $F\in \mathcal{F}, G\in \mathcal{G}$ and $\cap \{F\colon F\in…

The extremal problems regarding the maximum possible size of intersecting families of various combinatorial objects have been extensively studied. In this paper, we investigate supersaturation extensions, which in this context ask for the…

In this paper, we study the famous Erd\H{o}s--S\'os forbidden intersection problem for words over an alphabet of size $m$: what is the maximal size of a subfamily $\mathcal{F}$ of $[m]^n$ that does not contain two vectors $x, y$ coinciding…

The families $\mathcal{A}$ and $\mathcal{B}$ are cross intersecting if $A\cap B\ne \emptyset$ for any $A\in \mathcal{A}$ and $B\in \mathcal{B}$. Let $t\geq 2$ and $k_1\geq k_2\geq \cdots \geq k_t$. We say that $(\mathcal{F}_1, \dots,…

In this note, we give short proofs of three theorems about intersection problems. The first one is a determination of the maximum size of a nontrivial $k$-uniform, $d$-wise intersecting family for $n\ge \left(1+\frac{d}{2}\right)(k-d+2)$,…

A two-intersection set with parameters $(j;\alpha,\beta)$ for a block design is a $j$-subset of the point set of the design, which intersects every block in $\alpha$ or $\beta$ points. In this paper, we show the existence of a…

For any $\epsilon>0$ and $n>(1+\epsilon)t$, $n>n_0(\epsilon)$ we determine the size of the largest $t$-intersecting family of permutations, as well as give a sharp stability result. This resolves a conjecture of Ellis, Friedgut and Pilpel…

A family of sets is said to be intersecting if every pair of sets in the family have non-empty intersection. In this paper, we initiate the study of intersecting non-uniform families of sets of one of two sizes containing given subfamilies.…

A k-uniform linear cycle of length s is a cyclic list of k-sets A_1,..., A_s such that consecutive sets intersect in exactly one element and nonconsecutive sets are disjoint. For all k at least 5 and s at least 3 and sufficiently large n we…

In this work, we introduce a natural notion concerning finite vector spaces. A family of $k$-dimensional subspaces of $\mathbb{F}_q^n$, which forms a partial spread, is called almost affinely disjoint if any $(k+1)$-dimensional subspace…

We say that a set $A$ \emph{$t$-intersects} a set $B$ if $A$ and $B$ have at least $t$ common elements. Two families $\mathcal{A}$ and $\mathcal{B}$ are said to be \emph{cross-$t$-intersecting} if each set in $\mathcal{A}$ $t$-intersects…

This thesis generalizes the study of $C\cap(C + \alpha)$ where $C$ is the middle third Cantor set to self-affine sets in $\mathbb{R}^{n}$. We present sufficient and necessary conditions for when the translation $\alpha$ produces a…

This paper establishes an analog of the Erd\H{o}s-Ko-Rado theorem to polynomial rings over finite fields, affirmatively answering a conjecture of C. Tompkins. A $k$-uniform family of subsets of a set of finite size $n$ is $l$-intersecting…

Consider a system F of n polynomials in n variables, with a total of n+k distinct exponent vectors, over any local field L. We discuss conjecturally tight bounds on the maximal number of non-degenerate roots F can have over L, with all…

We prove that for $n$ sufficiently large, if $A$ is a family of permutations of $\{1,2,\ldots,n\}$ with no two permutations in $\mathcal{A}$ agreeing exactly once, then $|\mathcal{A}| \leq (n-2)!$, with equality holding only if…

Let $A$ and $B$ be sets in a finite vector space. In this paper, we study the magnitude of the set $A\cap f(B)$, where $f$ runs through a set of transformations. More precisely, we will focus on the cases that the set of transformations is…

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 prove the following variant of the Erd\H{o}s distinct subset sums problem. Given $t \ge 0$ and sufficiently large $n$, every $n$-element set $A$ whose subset sums are distinct modulo $N=2^n+t$ satisfies $$\max A \ge…

We give a characterization of the largest $2$-intersecting families of permutations of $\{1,2,\ldots,n\}$ and of perfect matchings of the complete graph $K_{2n}$ for all $n \geq 2$.

We show that any set $A$ in $\mathbb F_2^n$ with $|A+A| \le |A|^{2-\eta}$ must intersect a subspace of dimension $O_{\eta}(\log |A|)$ in at least $|A|^{\eta - o(1)}$ elements.