Related papers: Set systems without a 3-simplex
Let A be a subset of a finite abelian group G. We say that A is sum-free if there is no solution of the equation x + y = z, with x, y, z belonging to the set A. In this paper we shall characterise the largest possible sum-free subsets of G…
We study the maximum size of a subset of the $n \times n$ integer grid that does not contain specific geometric configurations, a variation of the classical problems initiated by Erd\H{o}s and Purdy. While extremal problems for 3-point…
A set of elements of a finite abelian group is called sum-free if it contains no Schur triple, i.e., no triple of elements $x,y,z$ with $x+y=z$. The study of how large the largest sum-free subset of a given abelian group is had started more…
How large can a family \cal A \subset \cal P [n] be if it does not contain A,B with |A\setminus B| = 1? Our aim in this paper is to show that any such family has size at most \frac{2+o(1)}{n} \binom {n}{\lfloor n/2\rfloor }. This is tight…
A cap set in $\mathbb{F}_3^n$ is a subset that contains no three elements adding to 0. Building on a construction of Edel, a recent paper of Tyrrell gave the first improvement to the lower bound for a size of a cap set in two decades…
Given a sequence of integers $a_j, j\ge 1,$ a multiset is a combinatorial object composed of unordered components, such that there are exactly $a_j$ one-component multisets of size $j.$ When $a_j\asymp j^{r-1} y^j$ for some $r>0$, $y\geq…
Let $3\le d\le k$ and $\nu\ge 0$ be fixed and $\mathcal{F}\subset\binom{[n]}{k}$. The matching number of $\mathcal{F}$, denoted by $\nu(\mathcal{F})$, is the maximum number of pairwise disjoint sets in $\mathcal{F}$, and $\mathcal{F}$ is…
We consider the problem of determining the maximum cardinality of a subset containing no arithmetic progressions of length $k$ in a given set of size $n$. It is proved that it is sufficient, in a certain sense, to consider the interval…
We consider families of k-subsets of the standard n-set. Two families F, G are said to be cross-intersecting if every member of F has non-empty intersection with every member of G. A family is called non-trivial if the intersection of all…
A finite set of real numbers is called convex if the differences between consecutive elements form a strictly increasing sequence. We show that, for any pair of convex sets $A, B\subset\mathbb R$, each of size $n^{1/2}$, the convex grid…
The well-known Sauer lemma states that a family $\mathcal{F}\subseteq 2^{[n]}$ of VC-dimension at most $d$ has size at most $\sum_{i=0}^d\binom{n}{i}$. We obtain both random and explicit constructions to prove that the corresponding…
In the paper we prove that any sumset or difference set has large E_3 energy. Also, we give a full description of families of sets having critical relations between some kind of energies such as E_k, T_k and Gowers norms. In particular, we…
We consider the following counting problem related to the card game SET: How many $k$-element SET-free sets are there in an $n$-dimensional SET deck? Through a series of algebraic reformulations and reinterpretations, we show the answer to…
For two families $\mathcal{A}, \mathcal{B} \subseteq \mathcal{P}([k])$, we write $\mathcal{A}\vdash\mathcal{B}$ if $A\not\supseteq B$ for each two sets $A \in \mathcal{A}$ and $B \in \mathcal{B}$. $\mathcal{A}$ and $\mathcal{B}$ are called…
The paper considers the problem of finding the largest possible set P(n), a subset of the set N of the natural numbers, with the property that a number is in P(n) if and only if it is a sum of n distinct naturals all in P(n) or none in…
Sidon sets are those sets such that the sums of two of its elements never coincide. They go back to the 30s when Sidon asked for the maximal size of a subset of consecutive integers with that property. This question is now answered in a…
A \textsl{Sperner $k$-partition system} on a set $X$ is a set of partitions of $X$ into $k$ classes such that the classes of the partitions form a Sperner set system (so no class from a partition is a subset of a class from another…
A set system is called union closed if for any two sets in the set system their union is also in the set system. Gilmer recently proved that in any union closed set system some element belongs to at least a $0.01$ fraction of sets, and…
Nonlinear complexity, as an important measure for assessing the randomness of sequences, is defined as the length of the shortest feedback shift registers that can generate a given sequence. In this paper, the structure of n-periodic binary…
The classical No-Three-In-Line problem seeks the maximum number of points that may be selected from an $n\times n$ grid while avoiding a collinear triple. The maximum is well known to be linear in $n$. Following a question of Erde, we seek…