Related papers: Moonflowers and efficient code sparsification

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…

A sunflower with p petals consists of p sets whose pairwise intersections are identical. The goal of the sunflower problem is to find the smallest r=r(p,k) such that any family of r^k distinct k-element sets contains a sunflower with p…

The famous Erd\H{o}s-Rado sunflower conjecture suggests that an $s$-sun\-flower-free family of $k$-element sets has size at most $(Cs)^k$ for some absolute constant $C$. In this note, we investigate the analog problem for $k$-spaces over…

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 $k$ 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.…

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…

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…

For a family $\mathcal{H} \subseteq \binom{[n]}{k}$, a subset $\{A_1, A_2, \ldots, A_m\} \subseteq \mathcal{H}$ is called a \textit{matching} of size~$m$ if the sets $A_1, A_2, \ldots, A_m$ are pairwise disjoint. The \textit{matching…

Robust sunflowers are a generalization of combinatorial sunflowers that have applications in monotone circuit complexity, DNF sparsification, randomness extractors, and recent advances on the Erd\H{o}s-Rado sunflower conjecture. The recent…

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 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…

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 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…

Extremal problems on set systems with restricted intersections have been an important part of combinatorics in the last 70 year. In this paper, we study the following Ramsey version of these problems. Given a set $L\subseteq…

We present some problems and results about variants of sunflowers in families of sets. In particular, we improve an upper bound of the first author, K\"orner and Monti on the maximum number of binary vectors of length $n$ so that every four…

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…

Fix an integer $k\ge 3$. Call a set $A\subseteq [N]$ LCM-$k$-free if it does not contain distinct $a_1,\dots,a_k$ such that $\mathrm{lcm}(a_i,a_j)$ is the same for all $1\le i<j\le k$. Define $$ f_k(N):=\max\left\{\sum_{a\in A}\frac1a:…

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…

A sunflower with $r$ petals is a collection of $r$ sets so that the intersection of each pair is equal to the intersection of all of them. Erd\H{o}s and Rado proved the sunflower lemma: for any fixed $r$, any family of sets of size $w$,…