Strongly intersecting integer partitions

We call a sum a1+a2+• • •+ak a partition of n of length k if a1, a2, . . . , ak and n are positive integers such that a1 ≤ a2 ≤ • • • ≤ ak and n = a1 + a2 + • • • + ak. For i = 1, 2, . . . , k, we call ai the ith part of the sum a1 + a2 + • • • + ak. Let Pn,k be the set of all partitions of n of length k. We say that two partitions a1+a2+• • •+ak and b1+b2+• • •+bk strongly intersect if ai = bi for some i. We call a subset A of Pn,k strongly intersecting if every two partitions in A strongly intersect. Let Pn,k(1) be the set of all partitions in Pn,k whose first part is 1. We prove that if 2 ≤ k ≤ n, then Pn,k(1) is a largest strongly intersecting subset of Pn,k, and uniquely so if and only if k ≥ 4 or k = 3 ≤ n ̸∈ {6, 7, 8} or k = 2 ≤ n ≤ 3.peer-reviewe

The Erdős-Ko-Rado properties of various graphs containing singletons

Let G=(V,E) be a graph. For r≥1, let be the family of independent vertex r-sets of G. For vV(G), let denote the star . G is said to be r-EKR if there exists vV(G) such that for any non-star family of pair-wise intersecting sets in . If the inequality is strict, then G is strictly r-EKR. Let Γ be the family of graphs that are disjoint unions of complete graphs, paths, cycles, including at least one singleton. Holroyd, Spencer and Talbot proved that, if GΓ and 2r is no larger than the number of connected components of G, then G is r-EKR. However, Holroyd and Talbot conjectured that, if G is any graph and 2r is no larger than μ(G), the size of a smallest maximal independent vertex set of G, then G is r-EKR, and strictly so if 2r<μ(G). We show that in fact, if GΓ and 2r is no larger than the independence number of G, then G is r-EKR; we do this by proving the result for all graphs that are in a suitable larger set Γ′Γ. We also confirm the conjecture for graphs in an even larger set Γ″Γ′

Cross-intersecting integer sequences

We call $(a_1, \dots, a_n)$ an \emph{$r$-partial sequence} if exactly $r$ of its entries are positive integers and the rest are all zero. For ${\bf c} = (c_1, \dots, c_n)$ with $1 \leq c_1 \leq \dots \leq c_n$, let $S_{\bf c}^{(r)}$ be the set of $r$-partial sequences $(a_1, \dots, a_n)$ with $0 \leq a_i \leq c_i$ for each $i$ in $\{1, \dots, n\}$, and let $S_{\bf c}^{(r)}(1)$ be the set of members of $S_{\bf c}^{(r)}$ which have $a_1 = 1$. We say that $(a_1, \dots, a_n)$ \emph{meets} $(b_1, \dots, b_m)$ if $a_i = b_i \neq 0$ for some $i$. Two sets $A$ and $B$ of sequences are said to be \emph{cross-intersecting} if each sequence in $A$ meets each sequence in $B$. Let ${\bf d} = (d_1, \dots, d_m)$ with $1 \leq d_1 \leq \dots \leq d_m$. Let $A \subseteq S_{\bf c}^{(r)}$ and $B \subseteq S_{\bf d}^{(s)}$ such that $A$ and $B$ are cross-intersecting. We show that $|A||B| \leq |S_{\bf c}^{(r)}(1)||S_{\bf d}^{(s)}(1)|$ if either $c_1 \geq 3$ and $d_1 \geq 3$ or ${\bf c} = {\bf d}$ and $r = s = n$. We also determine the cases of equality. We obtain this by proving a general cross-intersection theorem for \emph{weighted} sets. The bound generalises to one for $k \geq 2$ cross-intersecting sets.Comment: 20 pages, submitted for publication, presentation improve

Multiple cross-intersecting families of signed sets

A k-signed r-set on[n] = {1, ..., n} is an ordered pair (A, f), where A is an r-subset of [n] and f is a function from A to [k]. Families A1, ..., Ap are said to be cross-intersecting if any set in any family Ai intersects any set in any other family Aj. Hilton proved a sharp bound for the sum of sizes of cross-intersecting families of r-subsets of [n]. Our aim is to generalise Hilton's bound to one for families of k-signed r-sets on [n]. The main tool developed is an extension of Katona's cyclic permutation argument.peer-reviewe

An Intersection Theorem For Weighted Sets

. A weight function ! : 2 [n] ! R0 from the set of all subsets of [n] = f1; : : : ; ng to the nonnegative real numbers is called shift-- monotone in fm + 1; : : : ; ng if !(fi 1 ; : : : ; i m g) !(fj 1 ; : : : ; j m g) holds for all fi 1 ; : : : ; i m g, fj 1 ; : : : ; j m g ` [n] with i ` j ` ; ` = 1; : : : ; m, and if !(I) !(J) holds for all I ; J ` [n] with I ` J and J n I ` fm+1; : : : ; ng. A family F ` 2 [n] is called intersecting in [m] if F &quot; G &quot; [m] 6= ; for all F; G 2 F . Let !(F) = P F2F !(F ). We show that maxf!(F) : F ` 2 [n] ; F is intersecting in [n]g = maxf!(F) : F ` 2 [n] ; F is intersecting in [m]g provided that ! is shift--monotone in fm+ 1; : : : ; ng. An application to the poset of colored subsets of a finite set is given. Date: July 1998. 1. Introduction Let [n] = f1; 2; : : : ; ng, 2 [n] = fA ` [n]g and consider a weight function ! : 2 [n] ! R0 , where R0 denotes the set of all nonnegative real numbers. Define the weight of a family F ` ..