Related papers: The Oddtown problem modulo a composite number

A family $\mathcal A$ of subsets of an $n$-element set is called an eventown (resp. oddtown) if all its sets have even (resp. odd) size and all pairwise intersections have even size. Using tools from linear algebra, it was shown by…

Given a vector $\alpha = (\alpha_1, \ldots, \alpha_k) \in \mathbb{F}_2^k$, we say a collection of subsets $\mathcal{F}$ satisfies $\alpha$-intersection pattern modulo $2$ if all $i$-wise intersections consisting of $i$ distinct sets from…

Set systems with strongly restricted intersections, called $\alpha$-intersecting families for a vector $\alpha$, were introduced recently as a generalization of several well-studied intersecting families including the classical oddtown and…

Let $\mathcal{F}\subset 2^{[n]}$ be a set family such that the intersection of any two members of $\mathcal{F}$ has size divisible by $\ell$. The famous Eventown theorem states that if $\ell=2$ then $|\mathcal{F}|\leq 2^{\lfloor…

We study the supersaturation problems of oddtown and eventown. Given a family $\mathcal A$ of subsets of an $n$ element set, let $op(\mathcal A)$ denote the number of distinct pairs $A,B\in \mathcal A$ for which $|A \cap B|$ is odd. We show…

Given a collection $\mathcal{A}$ of subsets of an $n$ element set, let $\text{op}(\mathcal{A})$ denote the number of distinct pairs $A,B \in \mathcal{A}$ for which $|A \cap B|$ is odd. For $s \in \{1,2\}$, we prove $\text{op}(\mathcal{A})…

Let $G$ be the alternating group of degree $n$. Let $\omega(G)$ be the maximal size of a subset $S$ of $G$ such that $\langle x,y \rangle = G$ whenever $x,y \in S$ and $x \neq y$ and let $\sigma(G)$ be the minimal size of a family of proper…

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…

Let M be a subset of {0, .., n} and F be a family of subsets of an n element set such that the size of A intersection B is in M for every A, B in F. Suppose that l is the maximum number of consecutive integers contained in M and n is…

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…

It was asked by E. Szemer\'edi if, for a finite set $A\subset\mathbb{Z}$, one can improve estimates for $\max\{|A+A|,|A\cdot A|\}$, under the constraint that all integers involved have a bounded number of prime factors -- that is, each…

Let $\mathcal{A}_1,\ldots,\mathcal{A}_m$ be families of $k$-subsets of an $n$-set. Suppose that one cannot choose pairwise disjoint edges from $s+1$ distinct families. Subject to this condition we investigate the maximum of…

We give an overpartition analogue of Bressoud's combinatorial generalization of the G\"ollnitz-Gordon theorem for even moduli in general case. Let $\widetilde{O}_{k,i}(n)$ be the number of overpartitions of $n$ whose parts satisfy certain…

Let $\sigma(n)$ denote the sum of the positive divisors of $n$. We say that $n$ is perfect if $\sigma(n) = 2 n$. Currently there are no known odd perfect numbers. It is known that if an odd perfect number exists, then it must be of the form…

Suppose that $S \subseteq [n]^2$ contains no three points of the form $(x,y), (x,y+\delta), (x+\delta,y')$, where $\delta \neq 0$. How big can $S$ be? Trivially, $n \le |S| \le n^2$. Slight improvements on these bounds are obtained from…

We prove that every odd number $N$ greater than 1 can be expressed as the sum of at most five primes, improving the result of Ramar\'e that every even natural number can be expressed as the sum of at most six primes. We follow the circle…

A $k$-wise $\ell$-divisible set family is a collection $\mathcal{F}$ of subsets of ${ \{1,\ldots,n \} }$ such that any intersection of $k$ sets in $\mathcal{F}$ has cardinality divisible by $\ell$. If $k=\ell=2$, it is well-known that…

A family $\mathcal{F}\subseteq\mathcal{P}(n)$ is an $(a,b)$-town$\pmod k$ if all sets in it have cardinality $a\pmod k$ and all pairwise intersections in it have cardinality $b\pmod k$. For $k=2$ the maximal size of such a family is known…

Let $\omega(n)$ denote the number of distinct prime factors of $n$. Assuming a suitably uniform version of the prime $k$-tuples conjecture, we show that the number \begin{align*} \sum_{n=1}^\infty \frac{\omega(n)}{2^n} \end{align*} is…

Upper bounds to the size of a family of subsets of an n-element set that avoids certain configurations are proved. These forbidden configurations can be described by inclusion patterns and some sets having the same size. Our results are…