A Ramsey theory approach to ghostbusting

Biinfinite sequences X=(x_k)_k∈Z over the alphabet {0,1,...,q-1}, for an arbitrary q≥2, that satisfy the following q-ary ghost pulse (qGP) constraint: for all k,l,m∈Z such that x_k,x_l,x_m are nonzero and equal, x_k+l-m is also nonzero is studied in this paper. 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.