  A positive quadratic form is $(k,\ell)$-universal if it represents all the
numbers $kx+\ell$ where $x$ is a non-negative integer, and almost
$(k,\ell)$-universal if it represents all but finitely many of them. We prove
that for any $k,\ell$ such that $k\nmid\ell$ there exists an almost
$(k,\ell)$-universal diagonal ternary form. We also conjecture that there are
only finitely many primes $p$ for which a $(p,\ell)$-universal diagonal ternary
form exists (for any $\ell<p$) and we show the results of computer experiments
that speak in favor of the conjecture.
