Related papers: A note on $k$-wise oddtown problems

The directed Oberwolfach problem OP$^\ast(m_1,\ldots,m_k)$ asks whether the complete symmetric digraph $K_n^\ast$, assuming $n=m_1+\ldots +m_k$, admits a decomposition into spanning subdigraphs, each a disjoint union of $k$ directed cycles…

Fix an integer $r\ge2$. For each $n$ we consider families $\mathcal F\subseteq 2^{[n]}$ that form an antichain and have the property that, for every $t$, if there exists $A\in\mathcal F$ with $|A|=t$ then there exist at least $r$ members of…

Given integers $k,l\geq 2$, where either $l$ is odd or $k$ is even, let $n(k,l)$ denote the largest integer $n$ such that each element of $A_n$ is a product of $k$ many $l$-cycles. In 2008, M. Herzog, G. Kaplan and A. Lev proved that if…

We present a variety of new results on finite sets A of integers for which the sumset A+A is larger than the difference set A-A, so-called MSTD (more sums than differences) sets. First we show that there is, up to affine transformation, a…

For $k\geq i\geq 1$, let $B_{k,i}(n)$ denote the number of partitions of $n$ such that part 1 appears at most $i-1$ times, two consecutive integers l and $l+1$ appear at most $k-1$ times and if l and $l+1$ appear exactly $k-1$ times then…

Let $m(n)$ denote the maximum size of a family of subsets which does not contain two disjoint sets along with their union. In 1968 Kleitman proved that $m(n) = {n\choose m+1}+\ldots +{n\choose 2m+1}$ if $n=3m+1$. Confirming the conjecture…

Let $G$ be a simple graph with $n$ vertices and $m$ edges and let $k$ be a natural number such that $k\leq n.$ Brouwer conjectured that the sum of the $k$ largest Laplacian eigenvalues of $G$ is at most $m+{k+1 \choose 2}.$ In this paper we…

Let T_k^m={\sigma \in S_k | \sigma_1=m}. We prove that the number of permutations which avoid all patterns in T_k^m equals (k-2)!(k-1)^{n+1-k} for k <= n. We then prove that for any \tau in T_k^1 (or any \tau in T_k^k), the number of…

Let $\mathbb{Z}_n$ denote the ring of integers modulo $n$. In this paper we consider two extremal problems on permutations of $\mathbb{Z}_n$, namely, the maximum size of a collection of permutations such that the sum of any two distinct…

We show that whenever $s>k(k+1)$, then for any complex sequence $(\mathfrak a_n)_{n\in \mathbb Z}$, one has $$\int_{[0,1)^k}\left| \sum_{|n|\le N}\mathfrak a_ne(\alpha_1n+\ldots +\alpha_kn^k) \right|^{2s}\,{\rm d}{\mathbf \alpha}\ll…

There is no trivial mathematics, there are only trivial mathematicians! A mathematician is trivial if he or she believes that there exists trivial mathematics. Being a non-trivial mathematician myself, I will describe ten different proofs…

For integers $n\ge s\ge2$, let $e(n,s)$ denote the maximum size of a family $\F\subseteq2^{[n]}$ with no $s$ pairwise disjoint members. The problem of determining $e(n,s)$, now called the Erd\H{o}s--Kleitman problem, is the non-uniform…

We revisit the complexity of building, given a two-dimensional string of size $n$, an indexing structure that allows locating all $k$ occurrences of a two-dimensional pattern of size $m$. While a structure of size $\mathcal{O}(n)$ with…

We consider a continuous analogue of Babai et al.'s and Cai et al.'s problem of solving multiplicative matrix equations. Given $k+1$ square matrices $A_{1}, \ldots, A_{k}, C$, all of the same dimension, whose entries are real algebraic, we…

In this note we prove results of the following types. Let be given distinct complex numbers $z_j$ satisfying the conditions $|z_j| = 1, z_j \not= 1$ for $j=1,..., n$ and for every $z_j$ there exists an $ i$ such that $z_i = \bar{z_j}. $…

A $k$-partition of an $n$-set $X$ is a collection of $k$ pairwise disjoint non-empty subsets whose union is $X$. A family of $k$-partitions of $X$ is called $t$-intersecting if any two of its members share at least $t$ blocks. A…

We construct orthogonal arrays OA$_{\lambda} (k,n)$ (of strength two) having a row that is repeated $m$ times, where $m$ is as large as possible. In particular, we consider OAs where the ratio $m / \lambda$ is as large as possible; these…

We consider families of $k$-subsets of $\{1, \dots, n\}$, where $n$ is a multiple of $k$, which have no perfect matching. An equivalent condition for a family $\mathcal{F}$ to have no perfect matching is for there to be a blocking set,…

Let $A \subseteq F_2^n$ be a set with $|2A| = K|A|$. We prove that if (1) for at least a fraction $1-K^{-9}$ of all $s \in 2A$, the set $(A+s) \cap A$ has size at most $L\cdot|A|/K$, or (2) for at least a fraction $K^{-L}$ of all $s \in…

Generalizing a result (the case $k = 1$) due to M. A. Perles, we show that any polytopal upper bound sphere of odd dimension $2k + 1$ belongs to the generalized Walkup class ${\cal K}_k(2k + 1)$, i.e., all its vertex links are $k$-stacked…