Related papers: Dimension-Free Bounds for the Union-Closed Sets Co…

We prove that the conjecture made by Peter Frankl in the late 1970s is true. In other words for every finite union-closed family which contains a non?empty set, there is an element that belongs to at least half of its m

Let $r_5(N)$ be the largest cardinality of a set in $\{1,\ldots,N\}$ which does not contain $5$ elements in arithmetic progression. Then there exists a constant $c\in (0,1)$ such that \[r_5(N)\ll \frac{N}{\exp((\log\log N)^{c})}.\] Our work…

A celebrated unresolved conjecture of Peter Frankl states that every finite union-closed collection of sets ($B$), with non-empty universe, admits an abundant element. The best result in the literature states that if $|B|=n$, then there…

The famous Erdos-Heilbronn conjecture plays an important role in the development of additive combinatorics. In 2007 Z. W. Sun made the following further conjecture (which is the linear extension of the Erdos-Heilbronn conjecture): For any…

We prove that the union-closed sets conjecture is true for separating union-closed families $\mathcal{A}$ with $|\mathcal{A}| \leq 2\left(m+\frac{m}{\log_2(m)-\log_2\log_2(m)}\right)$ where $m$ denotes the number of elements in…

The Shapley-Folkman theorem shows that Minkowski averages of uniformly bounded sets tend to be convex when the number of terms in the sum becomes much larger than the ambient dimension. In optimization, Aubin and Ekeland [1976] show that…

Given $A$ a set of $N$ positive integers, an old question in additive combinatorics asks that whether $A$ contains a sum-free subset of size at least $N/3+\omega(N)$ for some increasing unbounded function $\omega$. The question is generally…

For a finite set $A\subset \mathbb{R}^d$, let $\Delta(A)$ denote the spread of $A$, which is the ratio of the maximum pairwise distance to the minimum pairwise distance. For a positive integer $n$, let $\gamma_d(n)$ denote the largest…

Let G be a graph with vertices V and edges E. Let F be the union-closed family of sets generated by E. Then F is the family of subsets of V without isolated points. Theorem: There is an edge e belongs to E such that |{U belongs to F | e…

A celebrated unit distance conjecture due to Erd\H os says that that the unit distances cannot arise more than $C_{\epsilon}n^{1+\epsilon}$ times (for any $\epsilon>0$) among $n$ points in the Euclidean plane (see e.g. \cite{SST84} and the…

In this paper we rectify two previous results found in the literature. Our work leads to a new upper bound for the largest sum-free subset of $[1,n]$ with lowest value in $\left [\frac{n}{3},\frac{n}{2}\right ]$, and the identification of…

A conjecture of Freiman gives an exact formula for the largest volume of a finite set $A$ of integers with given cardinality $k = |A|$ and doubling $T = |2A|$. The formula is known to hold when $T \le 3k-4$, for some small range over $3k-4$…

We derive a new estimate of the size of finite sets of points in metric spaces with few distances. The following applications are considered: (1) we improve the Ray-Chaudhuri--Wilson bound of the size of uniform intersecting families of…

Denote by $f_D(n)$ the maximum size of a set family $\mathcal{F}$ on $[n] \stackrel{\mbox{\normalfont\tiny def}}{=} \{1, \dots, n\}$ with distance set $D$. That is, $|A \bigtriangleup B| \in D$ holds for every pair of distinct sets $A, B…

In 1979 Frankl conjectured that in a finite non-trivial union-closed collection of sets there has to be an element that belongs to at least half the sets. We show that this is equivalent to the conjecture that in a finite non-trivial graph…

One of the prominent open problems in combinatorics is the discrepancy of set systems where each element lies in at most $t$ sets. The Beck-Fiala conjecture suggests that the right bound is $O(\sqrt{t})$, but for three decades the only…

A folklore conjecture asserts the existence of a constant $c_n > 0$ such that $\#\mathcal{F}_n(X) \sim c_n X$ as $X\to \infty$, where $\mathcal{F}_n(X)$ is the set of degree $n$ extensions $K/\mathbb{Q}$ with discriminant bounded by $X$.…

A set-system $S\subseteq \{0,1\}^n$ is cube-ideal if its convex hull can be described by capacity and generalized set covering inequalities. In this paper, we use combinatorics, convex geometry, and polyhedral theory to give exponential…

The $3k-4$ conjecture in groups $\mathbb{Z}/p\mathbb{Z}$ for $p$ prime states that if $A$ is a nonempty subset of $\mathbb{Z}/p\mathbb{Z}$ satisfying $2A\neq \mathbb{Z}/p\mathbb{Z}$ and $|2A|=2|A|+r \leq \min\{3|A|-4,\;p-r-4\}$, then $A$ is…

We improve both upper and lower bounds for the distribution-free testing of monotone conjunctions. Given oracle access to an unknown Boolean function $f:\{0,1\}^n \rightarrow \{0,1\}$ and sampling oracle access to an unknown distribution…