  This paper is concerned with games of infinite duration played over
potentially infinite graphs. Recently, Ohlmann (LICS 2022) presented a
characterisation of objectives admitting optimal positional strategies, by
means of universal graphs: an objective is positional if and only if it admits
well-ordered monotone universal graphs. We extend Ohlmann's characterisation to
encompass (finite or infinite) memory upper bounds.
  We prove that objectives admitting optimal strategies with
$\varepsilon$-memory less than $m$ (a memory that cannot be updated when
reading an $\varepsilon$-edge) are exactly those which admit well-founded
monotone universal graphs whose antichains have size bounded by $m$. We also
give a characterisation of chromatic memory by means of appropriate universal
structures. Our results apply to finite as well as infinite memory bounds (for
instance, to objectives with finite but unbounded memory, or with countable
memory strategies).
  We illustrate the applicability of our framework by carrying out a few case
studies, we provide examples witnessing limitations of our approach, and we
discuss general closure properties which follow from our results.
