A Ramsey theory approach to ghostbusting

-We study bi-infinite sequences x = (x(k))(kis an element ofZ) over the alphabet {0,1,...,q-1}, for an arbitrary q greater than or equal to 2, that satisfy the following q-ary ghost pulse (qGP) constraint: for all k,l, m is an element of Z such that x(k), x(l), x(m) are nonzero and equal, Xk+l-m is also nonzero. This constraint arises in the context of coding to combat the formation of spurious "ghost" pulses in high data-rate communication over an optical fiber. We show using techniques from Ramsey theory that if x satisfies the qGP constraint, then the support of x is a disjoint union of cosets of a subgroup kZ of Z and a set of zero density.