Two families $\mathcal{A}$ and $\mathcal{B}$ are cross-intersecting if $A\cap B\ne \emptyset$ for any $A\in \mathcal{A}$ and $B\in \mathcal{B}$. We call $t$ families $\mathcal{A}_1, \mathcal{A}_2,\dots, \mathcal{A}_t$ pairwise cross-intersecting families if $\mathcal{A}_i$ and $\mathcal{A}_j$ are cross-intersecting when $1\le i<j \le t$. Additionally, if $\mathcal{A}_j\ne \emptyset$ for each $j\in [t]$, then we say that $\mathcal{A}_1, \mathcal{A}_2,\dots, \mathcal{A}_t$ are non-empty pairwise cross-intersecting. Let $\mathcal{A}_1\subset{[n]\choose k_1}, \mathcal{A}_2\subset{[n]\choose k_2}, \dots, \mathcal{A}_t\subset{[n]\choose k_t}$ be non-empty pairwise cross-intersecting families with $t\geq 2$, $k_1\geq k_2\geq \cdots \geq k_t$, $n\ge k_1+k_2$ and $d_1, d_2, \dots, d_t$ be positive numbers. In this paper, we give a sharp upper bound of $\sum_{j=1}^td_j|\mathcal{A}_j|$ and characterize the families $\mathcal{A}_1, \mathcal{A}_2,\dots, \mathcal{A}_t$ attaining the upper bound. Our results unifies results of Frankl and Tokushige [J. Combin. Theory Ser. A 61 (1992)], Shi, Frankl and Qian [Combinatorica 42 (2022)], Huang and Peng \cite{huangpeng}, and Zhang-Feng \cite{ZF2023}. Furthermore, our result can be applied in the treatment for some $n<k_1+k_2$ while all previous known results do not have such an application. In the proof, a result of Kruskal-Katona is applied to allow us to consider only families $\mathcal{A}_i$ whose elements are the first $|\mathcal{A}_i|$ elements in lexicographic order. We bound $\sum_{i=1}^t{|\mathcal{A}_i|}$ by a single variable function $g(R)$, where $R$ is the last element of $\mathcal{A}_1$ in lexicographic order. One crucial and challenge part is to verify that $-g(R)$ has unimodality. We think that the unimodality of functions in this paper are interesting in their own, in addition to the extremal result.
We prove that given a finite collection of cylinders in $\R^3$ with the property that any two them intersect, then there is a line intersecting an $α$ fraction of the cylinders where $α=\frac 1{28}$. This is a special case of an interesting conjecture.
In this paper, we analyze the structure of maximal sets of $k$-dimensional spaces in $\mathrm{PG}(n,q)$ pairwise intersecting in at least a $(k-2)$-dimensional space, for $3 \leq k\leq n-2$. We give an overview of the largest examples of these sets with size more than $f(k,q)=\max\{3q^4+6q^3+5q^2+q+1,θ_{k+1}+q^4+2q^3+3q^2\}$.
Bernáth and Gerbner in 2007 introduced $(p,q)$-chain intersecting families of subsets of an $n$-element underlying set. Those have the property that for any $p$-chain $A_1\subsetneq A_2\subsetneq \dots \subsetneq A_p$ and $q$-chain $B_1\subsetneq B_2\subsetneq \dots \subsetneq B_q$, we have $A_p\cap B_q\neq \emptyset$. Bernáth and Gerbner determined the largest cardinality of such families. They also introduced strongly $(p,q)$-chain intersecting families, where $A_p\cap B_1\neq \emptyset$ and totally $(p,q)$-chain intersecting families, where $A_1\cap B_1\neq \emptyset$. They obtained some pa…
Let $2^{[n]}$ and $\binom{[n]}{i}$ be the power set and the class of all $i$-subsets of $\{1,2,\cdots,n\}$, respectively. We call two families $\mathscr{A}$ and $\mathscr{B}$ cross-intersecting if $A\cap B\neq \emptyset$ for any $A\in \mathscr{A}$ and $B\in \mathscr{B}$. In this paper we show that, for $n\geq k+l,l\geq r\geq 1,c>0$ and $\mathscr{A}\subseteq \binom{[n]}{k},\mathscr{B}\subseteq \binom{[n]}{l}$, if $\mathscr{A}$ and $\mathscr{B}$ are cross-intersecting and $\binom{n-r}{l-r}\leq|\mathscr{B}|\leq \binom{n-1}{l-1}$, then $$|\mathscr{A}|+c|\mathscr{B}|\leq \max\left\{\binom{n}{k}-\bi…
We prove the following the generalized Turán type result. A collection $\mathcal{T}$ of $r$ sets is an $r$-triangle if for every $T_1,T_2,\dots,T_{r-1}\in \mathcal{T}$ we have $\cap_{i=1}^{r-1}T_i\neq\emptyset$, but $\cap_{T\in \mathcal{T}}T$ is empty. A family $\mathcal{F}$ of sets is $r$-wise intersecting if for any $F_1,F_2,\dots,F_r\in \mathcal{F}$ we have $\cap_{i=1}^rF_i\neq \emptyset$ or equivalently if $\mathcal{F}$ does not contain any $m$-triangle for $m=2,3,\dots,r$. We prove that if $n\ge n_0(r,k)$, then the $r$-wise intersecting family $\mathcal{F}\subseteq \binom{[n]}{k}$ contain…
We consider families of k-subsets of the standard n-set. Two families F, G are said to be cross-intersecting if every member of F has non-empty intersection with every member of G. A family is called non-trivial if the intersection of all its members is empty. Supposing that F and G are non-trivial and cross-intersecting, we determine the maximum of |F|+|G|. For the proof a strengthened version of the so-called shifting technique is introduced. The corresponding problem for families of different uniformities is solved as well.
A $t$-intersecting constant dimension subspace code $C$ is a set of $k$-dimensional subspaces in a projective space PG(n,q), where distinct subspaces intersect in a $t$-dimensional subspace. A classical example of such a code is the sunflower, where all subspaces pass through the same $t$-space. The sunflower bound states that such a code is a sunflower if $|C| > \left( \frac {q^{k + 1} - q^{t + 1}}{q - 1} \right)^2 + \left( \frac {q^{k + 1} - q^{t + 1}}{q - 1} \right) + 1$. In this article we will look at the case $t=0$ and we will improve this bound for $q\geq 9$: a set $\mathcal{S}$ of $k$-…
For an integer $d \geq 2$, a family $\mathcal{F}$ of sets is $\textit{$d$-wise intersecting}$ if for any distinct sets $A_1,A_2,\dots,A_d \in \mathcal{F}$, $A_1 \cap A_2 \cap \dots \cap A_d \neq \emptyset$, and $\textit{non-trivial}$ if $\bigcap \mathcal{F} = \emptyset$. Hilton and Milner conjectured that for $k \geq d \geq 2$ and large enough $n$, the extremal non-trivial $d$-wise intersecting family of $k$-element subsets of $[n]$ is one of the following two families: \begin{align*} &\mathcal{H}(k,d) = \{A \in \binom{[n]}{k} : [d-1] \subset A, A \cap [d,k+1] \neq \emptyset\} \cup \{[k+1] \se…
We show that the maximum number of pairwise intersecting positive homothets of a $d$-dimensional centrally symmetric convex body, none of which contains the center of another in its interior, is at most $3^{d+1}$. Also, we improve upper bounds for cardinalities of $k$-distance sets in Minkowski spaces.
A set $A$ $t$-intersects a set $B$ if $A$ and $B$ have at least $t$ common elements. A set of sets is called a family. Two families $\mathcal{A}$ and $\mathcal{B}$ are cross-$t$-intersecting if each set in $\mathcal{A}$ $t$-intersects each set in $\mathcal{B}$. A family $\mathcal{H}$ is hereditary if for each set $A$ in $\mathcal{H}$, all the subsets of $A$ are in $\mathcal{H}$. The $r$th level of $\mathcal{H}$, denoted by $\mathcal{H}^{(r)}$, is the family of $r$-element sets in $\mathcal{H}$. A set $B$ in $\mathcal{H}$ is a base of $\mathcal{H}$ if for each set $A$ in $\mathcal{H}$, $B$ is n…
The number of k-intersections of an intersecting family of r-sets
Erdos-Ko-Rado定理告诉我们从一个n-集合中的r-集合的相交族可以有多大，而Lovasz和Tuza的结果给出了从这样一个族中的集合的成对相交可以出现的单例数的界限。我们认为这些问题的一个自然的概括。给定一个n-集的r-集的相交族和1个\leq k \leq r，有多少个k-集可以作为该族中的集的两两相交出现？对于k=r和k=1，这简化为上述问题。当n足够大时，我们对k和r的所有值都能准确地回答这个问题。我们还刻画了极值族。
Let $n > k > 1$ be integers, $[n] = \{1, \ldots, n\}$. Let $\mathcal F$ be a family of $k$-subsets of~$[n]$. The family $\mathcal F$ is called intersecting if $F \cap F' \neq \emptyset$ for all $F, F' \in \mathcal F$. It is called almost intersecting if it is not intersecting but to every $F \in \mathcal F$ there is at most one $F'\in \mathcal F$ satisfying $F \cap F' = \emptyset$. Gerbner et al. proved that if $n \geq 2k + 2$ then $|\mathcal F| \leq {n - 1\choose k - 1}$ holds for almost intersecting families. The main result implies the considerably stronger and best possible bound $|\mathca…
Let $\mathcal{A}\subseteq{[n]\choose a}$ and $\mathcal{B}\subseteq{[n]\choose b}$ be two families of subsets of $[n]$, we say $\mathcal{A}$ and $\mathcal{B}$ are cross-intersecting if $A\cap B\neq \emptyset$ for all $A\in\mathcal{A}$, $B\in\mathcal{B}$. In this paper, we study cross-intersecting families in the multi-part setting. By characterizing the independent sets of vertex-transitive graphs and their direct products, we determine the sizes and structures of maximum-sized multi-part cross-intersecting families. This generalizes the results of Hilton's and Frankl--Tohushige's on cross-inte…
The notion of cross intersecting set pair system of size $m$, $\Big(\{A_i\}_{i=1}^m, \{B_i\}_{i=1}^m\Big)$ with $A_i\cap B_i=\emptyset$ and $A_i\cap B_j\ne\emptyset$, was introduced by Bollobás and it became an important tool of extremal combinatorics. His classical result states that $m\le {a+b\choose a}$ if $|A_i|\le a$ and $|B_i|\le b$ for each $i$. Our central problem is to see how this bound changes with the additional condition $|A_i\cap B_j|=1$ for $i\ne j$. Such a system is called $1$-cross intersecting. We show that the maximum size of a $1$-cross intersecting set pair system is -- at…
A convex polygon $A$ is related to a convex $m$-gon $K= \bigcap_{i=1}^m k_i^+$, where $k_1^+,..., k_m^+$ are the $m$ halfplanes whose intersection is equal to $K$, if $A$ is the intersection of halfplanes $a_1^+,...,a_l$, each of which is a translate of one of the $k_i^+$-s. The planar family ${\cal A}$ is related to $K$ if each $A \in {\cal A}$ is related to $K$. We prove that any family of pairwise intersecting convex sets related to a given $n$-gon has a finite piercing number which depends on $n$. In the general case we show $O(3^{n^3})$, while for a certain class of families, we decrease …
Families $\mathcal{A}_1, \mathcal{A}_2, ..., \mathcal{A}_k$ of sets are said to be \emph{cross-intersecting} if for any $i$ and $j$ in $\{1, 2, ..., k\}$ with $i \neq j$, any set in $\mathcal{A}_i$ intersects any set in $\mathcal{A}_j$. For a finite set $X$, let $2^X$ denote the \emph{power set of $X$} (the family of all subsets of $X$). A family $\mathcal{H}$ is said to be \emph{hereditary} if all subsets of any set in $\mathcal{H}$ are in $\mathcal{H}$; so $\mathcal{H}$ is hereditary if and only if it is a union of power sets. We conjecture that for any non-empty hereditary sub-family $\math…
Two sets $\mathscr{A}$ and $\mathscr{B}$ are said to be cross-intersecting if $X\cap Y\neq\emptyset$ for all $X\in\mathscr{A}$ and $Y\in\mathscr{B}$. Given two cross-intersecting Sperner families (or antichains) $\mathscr{A}$ and $\mathscr{B}$ of $\mathbb{N}_n$, we prove that $|\mathscr{A}|+|\mathscr{B}|\le 2{{n}\choose{\lceil{n/2}\rceil}}$ if $n$ is odd, and $|\mathscr{A}|+|\mathscr{B}|\le {{n}\choose{n/2}}+{{n}\choose{(n/2)+1}}$ if $n$ is even. Furthermore, all extremal and almost-extremal families for $\mathscr{A}$ and $\mathscr{B}$ are determined.
A family $\mathcal{F}$ of subsets of $\{1,\dots,n\}$ is called $k$-wise intersecting if any $k$ members of $\mathcal{F}$ have non-empty intersection, and it is called maximal $k$-wise intersecting if no family strictly containing $\mathcal{F}$ satisfies this condition. We show that for each $k\geq 2$ there is a maximal $k$-wise intersecting family of size $O(2^{n/(k-1)})$. Up to a constant factor, this matches the best known lower bound, and answers an old question of Erdős and Kleitman, recently studied by Hendrey, Lund, Tompkins, and Tran.
Two families $\mathcal{F},\mathcal{G}$ of $k$-subsets of $\{1,2,\ldots,n\}$ are called non-trivial cross-intersecting if $F\cap G\neq \emptyset$ for all $F\in \mathcal{F}, G\in \mathcal{G}$ and $\cap \{F\colon F\in \mathcal{F}\}=\emptyset=\cap \{G\colon G\in \mathcal{G}\}$. In the present paper, we determine the maximum product of the sizes of two non-trivial cross-intersecting families of $k$-subsets of $\{1,2,\ldots,n\}$ for $n\geq 4k$, $k\geq 8$, which is a product version of the classical Hilton-Milner Theorem.
School of Mathematics and Statistics, HNP-LAMA, Central South University（中南大学数学与统计学院）；Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）；School of Mathematics and Statistics, Qinghai Minzu University（青海民族大学数学与统计学院）；Qinghai Institute of Applied Mathematics（青海省应用数学研究所）
研究非均匀 t-相交族的上界，通过生成集方法和推拉方法证明了多个族的交叉 t-相交条件下的最大规模限制，并扩展了 Katona 和 Frankl 等人的结果。
Let $k\ge d\ge 3$ be fixed. Let $\mathcal{F}$ be a $k$-uniform family on $[n]$. Then $\mathcal{F}$ is $(d,s)$-conditionally intersecting if it does not contain $d$ sets with union of size at most $s$ and empty intersection. Answering a question of Frankl, we present some structural results for families that are $(d,s)$-conditionally intersecting with $s\ge 2k+d-3$, and families that are $(k,2k)$-conditionally intersecting. As applications of our structural results, we present some new proofs to the upper bounds for the size of the following $k$-uniform families on $[n]$. (a) $(d,2k+d-3)$-condi…
Two hypergraphs $H_1,\ H_2$ are called {\em cross-intersecting} if $e_1 \cap e_2 \neq \emptyset$ for every pair of edges $e_1 \in H_1,~e_2 \in H_2$. Each of the hypergraphs is then said to {\em block} the other. Given parameters $n,r,m$ we determine the maximal size of a sub-hypergraph of $[n]^r$ (meaning that it is $r$-partite, with all sides of size $n$) for which there exists a blocking sub-hypergraph of $[n]^r$ of size $m$. The answer involves a fractal-like (that is, self-similar) sequence, first studied by Knuth. We also study the same question with $\binom{n}{r}$ replacing $[n]^r$.
We prove some properties of the Kruskal-Katona function, and apply to the following variation of cross-intersecting antichains. Let $n\ge 4$ be an even integer and $\mathscr{A}$ and $\mathscr{B}$ be two cross-intersecting antichains of $\mathbb{N}_n$ with at most $k$ disjoint pairs, i.e. for all $A_i\in \mathscr{A}$, $B_j\in\mathscr{B}$, $A_i\cap B_j=\emptyset$ only if $i=j\le k$. We prove a best possible upper bound on $|\mathscr{A}|+|\mathscr{B}|$. Furthermore, we show that the extremal families contain only $\frac{n}{2}$ and $(\frac{n}{2}+1)$-sets.
A family of sets is said to be intersecting if every pair of sets in the family have non-empty intersection. In this paper, we initiate the study of intersecting non-uniform families of sets of one of two sizes containing given subfamilies. For a set $X$ and integer $r$, let $\binom{X}{r}$ denote the family $\{A \subseteq X: |X| = r\}$. Let $a$, $b$, and $n$ be positive integers such that $a < b$. We determine the maximum size of an intersecting family in $\binom{[n]}{a} \cup \binom{[2n]}{b}$ whenever $n > b$. For $n$ sufficiently large, we also determine the maximum size of an intersecting fa…
A well-known result of Bollobás says that if $\{(A_i, B_i)\}_{i=1}^m$ is a set pair system such that $|A_i| \le a$ and $|B_i| \le b$ for $1 \le i \le m$, and $A_i \cap B_j \ne \emptyset$ if and only if $i \ne j$, then $m \le {a+b \choose a}$. Füredi, Gyárfás and Király recently initiated the study of such systems with the additional property that $|A_i \cap B_j| = 1$ for all $i \ne j$. Confirming a conjecture of theirs, we show that this extra condition allows an improvement of the upper bound (at least) by a constant factor.
For positive integers $n$ and $r$ such that $r \leq \lfloor n/2\rfloor$, let $X$ be a set of $n$ elements and let $\binom{X}{r}$ be the family of all $r$-subsets of $X$. Two sub-families $\mathcal{A}$ and $\mathcal{B}$ of $\binom{X}{r}$ are called cross intersecting if $A \cap B \neq \emptyset$ for all $A \in \mathcal{A}$ and $B \in \mathcal{B}$. One of main tools in the study of extremal set theory, and cross intersecting families in particular, is compression operation. In this paper, we give an example of cross intersecting families $\mathcal{A}$ and $\mathcal{B}$ that the compression opera…
A family $\mathcal{F}$ on ground set $[n]:=\{1,2,\ldots, n\}$ is maximal $k$-wise intersecting if every collection of at most $k$ sets in $\mathcal{F}$ has non-empty intersection, and no other set can be added to $\mathcal{F}$ while maintaining this property. In 1974, Erdős and Kleitman asked for the minimum size of a maximal $k$-wise intersecting family. We answer their question for $k=3$ and sufficiently large $n$. We show that the unique minimum family is obtained by partitioning the ground set $[n]$ into two sets $A$ and $B$ with almost equal sizes and taking the family consisting of all t…
A set of sets is called a family. Two families $\mathcal{A}$ and $\mathcal{B}$ of sets are said to be cross-intersecting if each member of $\mathcal{A}$ intersects each member of $\mathcal{B}$. For any two integers $n$ and $k$ with $1 \leq k \leq n$, let ${[n] \choose \leq k}$ denote the family of subsets of $[n] = \{1, \dots, n\}$ that have at most $k$ elements. We show that if $\mathcal{A}$ is a non-empty subfamily of ${[n] \choose \leq r}$, $\mathcal{B}$ is a non-empty subfamily of ${[n] \choose \leq s}$, $r \leq s$, and $\mathcal{A}$ and $\mathcal{B}$ are cross-intersecting, then \[|\mathc…
A family $\mathcal{F}$ on ground set $\{1,2,\ldots, n\}$ is maximal $k$-wise intersecting if every collection of $k$ sets in $\mathcal{F}$ has non-empty intersection, and no other set can be added to $\mathcal{F}$ while maintaining this property. Erdős and Kleitman asked for the minimum size of a maximal $k$-wise intersecting family. Complementing earlier work of Hendrey, Lund, Tompkins and Tran, who answered this question for $k=3$ and large even $n$, we answer it for $k=3$ and large odd $n$. We show that the unique minimum family is obtained by partitioning the ground set into two sets $A$ a…
In this paper we present a simple unifying approach to prove several statements about intersecting and cross-intersecting families, including the Erd\H os--Ko--Rado theorem, the Hilton--Milner theorem, a theorem due to Frankl concerning the size of intersecting families with bounded maximal degree, and versions of results on the sum of sizes of non-empty cross-intersecting families due to Frankl and Tokushige. Several new stronger results are also obtained. Our approach is based on the use of regular bipartite graphs. These graphs are quite often used in Extremal Set Theory problems, however, …
For an $n$-element set $X$ let $\binom{X}{k}$ be the collection of all its $k$-subsets. Two families of sets $\mathcal A$ and $\mathcal B$ are called cross-intersecting if $A\cap B \neq \emptyset$ holds for all $A\in\mathcal A$, $B\in\mathcal B$. Let $f(n,k)$ denote the maximum of $\min\{|\mathcal A|, |\mathcal B|\}$ where the maximum is taken over all pairs of {\em disjoint}, cross-intersecting families $\mathcal A, \mathcal B\subset\binom{[n]}{k}$. Let $c=\log_2e$. We prove that $f(n,k)=\left\lfloor\frac12\binom{n-1}{k-1}\right\rfloor$ essentially iff $n>ck^2$ (cf. Theorem~1.4 for the exact …
A pseudocircle is a simple closed curve on the sphere or in the plane. The study of arrangements of pseudocircles was initiated by Grünbaum, who defined them as collections of simple closed curves that pairwise intersect in exactly two crossings. Grünbaum conjectured that the number of triangular cells $p_3$ in digon-free arrangements of $n$ pairwise intersecting pseudocircles is at least $2n-4$. We present examples to disprove this conjecture. With a recursive construction based on an example with $12$ pseudocircles and $16$ triangles we obtain a family with $p_3(\mathcal{A})/n \to 16/11 = 1.…
A family of subsets of $\{1,\ldots,n\}$ is called {\it intersecting} if any two of its sets intersect. A classical result in extremal combinatorics due to Erdős, Ko, and Rado determines the maximum size of an intersecting family of $k$-subsets of $\{1,\ldots, n\}$. In this paper we study the following problem: how many intersecting families of $k$-subsets of $\{1,\ldots, n\}$ are there? Improving a result of Balogh, Das, Delcourt, Liu, and Sharifzadeh, we determine this quantity asymptotically for $n\ge 2k+2+2\sqrt{k\log k}$ and $k\to \infty$. Moreover, under the same assumptions we also deter…
A long standing conjecture of Richter and Thomassen states that the total number of intersection points between any $n$ simple closed Jordan curves in the plane, so that any pair of them intersect and no three curves pass through the same point, is at least $(1-o(1))n^2$. We confirm the above conjecture in several important cases, including the case (1) when all curves are convex, and (2) when the family of curves can be partitioned into two equal classes such that each curve from the first class is touching every curve from the second class. (Two curves are said to be touching if they have pr…
The celebrated Erdős-Ko-Rado theorem \cite{EKR1961} states that the maximum intersecting $k$-uniform family on $[n]$ is a full star if $n\ge 2k+1$. Furthermore, Hilton-Milner \cite{HM1967} showed that if an intersecting $k$-uniform family on $[n]$ is not a subfamily of a full star, then its maximum size achieves only on a family isomorphic to $HM(n,k):= \Bigl\{G\in {[n] \choose k}: 1\in G, G\cap [2,k+1] \neq \emptyset \Bigr\} \cup \Bigl\{ [2,k+1] \Bigr\} $ if $n>2k$ and $k\ge 4$, and there is one more possibility in the case of $k=3$. Han and Kohayakawa \cite{HK2017} determined the maximum int…
In this paper we investigate Erdős-Ko-Rado theorems in ovoidal circle geometries. We prove that in Möbius planes of even order greater than 2, and ovoidal Laguerre planes of odd order, the largest families of circles which pairwise intersect in at least one point, consist of all circles through a fixed point. In ovoidal Laguerre planes of even order, a similar result holds, but there is one other type of largest family of pairwise intersecting circles. As a corollary, we prove that the largest families of polynomials over $\mathbb F_q$ of degree at most $k$, with $2 \leq k < q$, which pairwise…
Let $S$ be an $n$-punctured sphere, with $n \geq 3$. We prove that $\binom{n}{3}$ is the maximum size of a family of pairwise non-homotopic simple arcs on $S$ joining a fixed pair of distinct punctures of $S$ and pairwise intersecting at most twice. On the way, we show that a square annular diagram $A$ has a corner on each of its boundary paths if $A$ contains at least one square and the dual curves of $A$ are simple arcs joining the boundary paths of $A$ and pairwise intersecting at most once.
A family $\mathcal{A}$ of sets is said to be intersecting if every two sets in $\mathcal{A}$ intersect. Two families $\mathcal{A}$ and $\mathcal{B}$ are said to be cross-intersecting if each set in $\mathcal{A}$ intersects each set in $\mathcal{B}$. For a positive integer $n$, let $[n] = \{1, \dots, n\}$ and $\mathcal{S}_n = \{A \subseteq [n] \colon 1 \in A\}$. In this note, we extend the Erdős-Ko-Rado Theorem by showing that if $\mathcal{A}$ and $\mathcal{B}$ are non-empty cross-intersecting families of subsets of $[n]$, $\mathcal{A}$ is intersecting, and $a_0, a_1, \dots, a_n, b_0, b_1, \dot…
For a planar point-set $P$, let D(P) be the minimum number of pairwise-disjoint empty disks such that each point in $P$ lies on the boundary of some disk. Further define D(n) as the maximum of D(P) over all n-element point sets. Hosono and Urabe recently conjectured that $D(n)=\lceil n/2 \rceil$. Here we show that $D(n) \geq n/2 + n/236 - O(\sqrt{n})$ and thereby disprove this conjecture.
