Related papers: Sunflowers of Convex Open Sets

A sunflower is a family of sets that have the same pairwise intersections. We simplify a recent result of Alweiss, Lovett, Wu and Zhang that gives an upper bound on the size of every family of sets of size $k$ that does not contain a…

Sunflowers, or $\Delta$-systems, are a fundamental concept in combinatorics introduced by Erd\H{o}s and Rado in their paper: {\em Intersection theorems for systems of sets}, J. Lond. Math. Soc. (1) {\bf 35} (1960), 85--90. A sunflower is a…

A sunflower with $r$ petals is a collection of $r$ sets over a ground set $X$ such that every element in $X$ is in no set, every set, or exactly one set. Erd\H{o}s and Rado \cite{er} showed that a family of sets of size $n$ contains a…

A sunflower is a collection of distinct sets such that the intersection of any two of them is the same as the common intersection $C$ of all of them, and $|C|$ is smaller than each of the sets. A longstanding conjecture due to Erd\H{o}s and…

A collection of distinct sets is called a sunflower if the intersection of any pair of sets equals the common intersection of all the sets. Sunflowers are fundamental objects in extremal set theory with relations and applications to many…

A family of $r$ distinct sets $\{A_1,\ldots, A_r\}$ is an $r$-sunflower if for all $1 \leqslant i < j \leqslant r$ and $1 \leqslant i' < j' \leqslant r$, we have $A_i \cap A_j = A_{i'} \cap A_{j'}$. Erd\H{o}s and Rado conjectured in 1960…

We study sunflowers within the context of finitely generated substructures of ultrahomogeneous structures. In particular, we look at bounds on how large a set system is needed to guarantee the existence of sunflowers of a given size. We…

Let $B_{x}\subseteq\mathbb{R}^{n}$ denote the Euclidean ball with diameter $[0,x]$, i.e. with with center at $\frac{x}{2}$ and radius $\frac{\left|x\right|}{2}$. We call such a ball a petal. A flower $F$ is any union of petals, i.e.…

A uniform hypergraph is called a sunflower if all of its hyperedges intersect in the same set of vertices. In this paper, we determine the eigenvalues and spectral moments of a sunflower, thereby obtaining an explicit formula for its…

Let $f(k,r,s)$ stand for the least number so that if $\cal F$ is an arbitrary $k$-uniform, $L$-intersecting set system, where $|L|=s$, and $\cal F$ has more than $f(k,r,s)$ elements, then $\cal F$ contains a sunflower with $r$ petals. We…

A $h$-sunflower in a hypergraph is a family of edges with $h$ vertices in common. We show that if we colour the edges of a complete hypergraph in such a way that any monochromatic $h$-sunflower has at most $\lambda$ petals, then it contains…

Given a family $\mathcal F$ of $k$-element sets, $S_1,\ldots,S_r\in\mathcal F$ form an {\em $r$-sunflower} if $S_i \cap S_j =S_{i'} \cap S_{j'}$ for all $i \neq j$ and $i' \neq j'$. According to a famous conjecture of Erd\H os and Rado…

A collection of $k$ sets is said to form a $k$-sunflower, or $\Delta$-system, if the intersection of any two sets from the collection is the same, and we call a family of sets $\mathcal{F}$ sunflower-free if it contains no sunflowers.…

Extending the notion of sunflowers, we call a family of at least two sets an odd-sunflower if every element of the underlying set is contained in an odd number of sets or in none of them. It follows from the Erd\H os--Szemer\'edi…

This paper explores the structure of the combinatorial domain $2^X$ in relation to sunflowers. The previous study found some intrinsic properties of the $l$-extension \[ Ext \left( \mathcal{F}, l \right) = \left\{ V ~:~ V \in {X \choose…

We demonstrate the truth of the sunflower conjecture by showing that a family $\mathcal{F}$ of sets each of cardinality at most $m$ includes a $k$-sunflower, if $|\mathcal{F}| > ( c k )^{2m}$ for a constant $c>0$ independent of $m$ and $k$,…

We prove the following Helly-type result. Let $\mathcal{C}_1,\dots,\mathcal{C}_{3d}$ be finite families of convex bodies in $\mathbb{R}^d$. Assume that for any colorful selection of $2d$ sets, $C_{i_k}\in \mathcal{C}_{i_k}$ for each $1\leq…

The Union Closed Sets Conjecture is one of the most renowned problems in combinatorics. Its appeal lies in the simplicity of its statement contrasted with the potential complexity of its resolution. The conjecture posits that, in any union…

A sunflower with $k$ petals, or $k$-sunflower, is a family of $k$ sets every two of which have a common intersection. Known since 1960, the sunflower conjecture states that a family ${\mathcal F}$ of sets each of cardinality $m$ includes a…

The Erd\H{o}s--Rado sunflower problem admits two natural analogues in finite vector spaces, corresponding to two different ways of generalising the set-theoretic notion of a sunflower. The first, used by Ihringer and Kupavskii [FFA 110…