We call (a1β,β¦,anβ) an \emph{r-partial sequence} if exactly r of
its entries are positive integers and the rest are all zero. For c=(c1β,β¦,cnβ) with 1β€c1ββ€β―β€cnβ, let Sc(r)β
be the set of r-partial sequences (a1β,β¦,anβ) with 0β€aiββ€ciβ for each i in {1,β¦,n}, and let Sc(r)β(1) be the set
of members of Sc(r)β which have a1β=1. We say that (a1β,β¦,anβ) \emph{meets} (b1β,β¦,bmβ) if aiβ=biβξ =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 d=(d1β,β¦,dmβ)
with 1β€d1ββ€β―β€dmβ. Let AβSc(r)β and BβSd(s)β such that A and B are cross-intersecting. We
show that β£Aβ£β£Bβ£β€β£Sc(r)β(1)β£β£Sd(s)β(1)β£ if either c1ββ₯3 and d1ββ₯3 or c=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 kβ₯2 cross-intersecting sets.Comment: 20 pages, submitted for publication, presentation improve