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• W2282951856 abstract "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." @default.
• W2282951856 created "2016-06-24" @default.
• W2282951856 creator A5038574319 @default.
• W2282951856 date "2012-12-31" @default.
• W2282951856 modified "2023-09-27" @default.
• W2282951856 title "Cross-intersecting integer sequences" @default.
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