相关论文: Delta-system method: a survey
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…
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…
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 call a family of $s$ sets $\{F_1, \ldots, F_s\}$ a \textit{sunflower with $s$ petals} if, for any distinct $i, j \in [s]$, one has $F_i \cap F_j = \cap_{u = 1}^s F_u$. The set $C = \cap_{u = 1}^s F_u$ is called the {\it core} of the…
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…
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…
A collection of sets is {\em intersecting} if every two members have nonempty intersection. We describe the structure of intersecting families of $r$-sets of an $n$-set whose size is quite a bit smaller than the maximum ${n-1 \choose r-1}$…
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…
The study of intersection problems in Extremal Combinatorics dates back perhaps to 1938, when Paul Erd\H{o}s, Chao Ko and Richard Rado proved the (first) `Erd\H{o}s-Ko-Rado theorem' on the maximum possible size of an intersecting family of…
Celebrated results often unfold like episodes in a long-running series. In the field of extremal set thoery, Erd\H{o}s, Ko, and Rado in 1961 established that any $k$-uniform intersecting family on $[n]$ has a maximum size of…
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…
Many classical theorems in combinatorics establish the emergence of substructures within sufficiently large collections of objects. Well-known examples are Ramsey's theorem on monochromatic subgraphs and the Erd\H{o}s-Rado sunflower lemma.…
We develop a new approach to approximate families of sets, complementing the existing `$\Delta$-system method' and `junta approximations method'. The approach, which we refer to as `spread approximations method', is based on the notion 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…
Well-graded families, extremal systems and maximum systems (the last two in the sense of VC-theory and Sauer-Shelah lemma on VC-dimension) are three important classes of set systems. This paper aims to study the notion of duality in the…
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…
Given a root system $R$, two roots are said to be \emph{strongly orthogonal} if neither their sum nor difference is a root. Gashi defined a family of graphs with vertices labelled by sums of $k$-element strongly orthogonal subsets of roots,…
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…