Related papers: Sunflowers in set systems with small VC-dimension

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…

We prove that given a constant $k \ge 2$ and a large set system $\mathcal{F}$ of sets of size at most $w$, a typical $k$-tuple of sets $(S_1, \cdots, S_k)$ from $\mathcal{F}$ can be ``blown up" in the following sense: for each $1 \le i \le…

In 1977, Duke and Erd\H{o}s asked the following general question: What is the largest size of a family $\mathtt{F} \subset \binom{[n]}{k}$ that does not contain a sunflower with $s$ petals and core of size exactly $t - 1$? This problem is…

A sunflower with a core $Y$ is a family ${\cal B}$ of sets such that $U \cap U' = Y$ for each two different elements $U$ and $U'$ in ${\cal B}$. The well-known sunflower lemma states that a given family ${\cal F}$ of sets, each of…

A family of independent $r$-sets of a graph $G$ is an $r$-star if every set in the family contains some fixed vertex $v$. A graph is $r$-EKR if the maximum size of an intersecting family of independent $r$-sets is the size of an $r$-star.…

We show that a family $\mathcal{F}$ of sets each of cardinality $m \in \mathbb{Z}_{>2}$ includes a $k$-sunflower if $ |\mathcal{F}| \ge \left( \frac{c k^2 \ln m}{\ln \ln m} \right)^m$ for some constant $c>0$, where $k$-sunflower means a…

A family of $k$-subsets $A_1, A_2, ..., A_d$ on $[n]=\{1,2,..., n\}$ is called a $(d, c)$-cluster if the union $A_1\cup A_2 \cup ... \cup A_d$ contains at most $ck$ elements with $c<d$. Let $\mathcal{F}$ be a family of $k$-subsets of an…

An $r$-sunflower is a collection of $r$ sets such that the intersection of any two sets in the collection is identical. We analyze a random process which constructs a $w$-uniform $r$-sunflower free family starting with an empty family and…

We combine here Tao's slice-rank bounding method and Gr\"obner basis techniques and apply here to the Erd\H{o}s-Rado Sunflower Conjecture. Let $\frac{3k}{2}\leq n\leq 3k$ be integers. We prove that if $\mbox{$\cal F$}$ be a $k$-uniform…

A family of sets is $s$-intersecting if every pair of its sets has at least $s$ elements in common. It is an $s$-star if all its members have some $s$ elements in common. A family of sets is called $s$-EKR if all its $s$-intersecting…

A family of sets is intersecting if every pair of its sets intersect. A star is a family with some element (a center) in each of its sets. The classical 1961 result of Erd\H{o}s, Ko, and Rado states that every intersecting family of r-sets…

We introduce \emph{moonflowers}, a weaker analogue of sunflowers. A family of sets $S_1,\ldots,S_k$ is a $k$-moonflower if each set $S_i$ contains at least one element that is absent from all the others. We study the extremal problem of…

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…

Frankl--Pach and Erd\H{o}s conjectured that any $(d+1)$-uniform set family $\mathcal{F}\subseteq \binom{[n]}{d+1}$ with VC-dimension at most $d$ has size at most $\binom{n-1}{d}$ when $n$ is sufficiently large. Ahlswede and Khachatrian…

Alon, Shpilka and Umans considered the following version of usual sunflower-free subset: a subset $\mbox{$\cal F$}\subseteq \{1,\ldots ,D\}^n$ for $D>2$ is sunflower-free if for every distinct triple $x,y,z\in \mbox{$\cal F$}$ there exists…

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…

Given a graph $G$, let $f_{G}(n,m)$ be the minimal number $k$ such that every $k$ independent $n$-sets in $G$ have a rainbow $m$-set. Let $\mathcal{D}(2)$ be the family of all graphs with maximum degree at most two. Aharoni et al. (2019)…

A $d$-simplex is defined to be a collection $A_1,\dots,A_{d+1}$ of subsets of size $k$ of $[n]$ such that the intersection of all of them is empty, but the intersection of any $d$ of them is non-empty. Furthermore, a $d$-cluster is a…

Let $\mathcal{A}$ be a family of subsets of a finite set. A subfamily of $\mathcal{A}$ is said to be intersecting when any two of its members contain at least one common element. We say that $\mathcal{A}$ is an Erd{\H o}s-Ko-Rado (EKR)…

A family of sets is intersecting if any two sets in the family intersect. Given a graph $G$ and an integer $r\geq 1$, let $\mathcal{I}^{(r)}(G)$ denote the family of independent sets of size $r$ of $G$. For a vertex $v$ of $G$, let…