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.

翻译：本文关注在潜在无限图上进行的无限持续时间博弈。近期，Ohlmann（LICS 2022）通过通用图理论提出了对允许最优位置策略的目标的刻画：一个目标是位置性的当且仅当它允许良序单调通用图。我们将Ohlmann的刻画推广至（有限或无限）记忆上界情形。我们证明：允许ε-记忆小于m（一种在读取ε-边时无法更新的记忆）的最优策略的目标，正是那些允许良基单调通用图且其反链大小以m为界的目标。我们还通过适当的通用结构给出了色记忆的刻画。我们的结论适用于有限及无限记忆界（例如，具有有限但无界记忆的目标，或具有可数记忆策略的目标）。我们通过若干案例研究展示该框架的适用性，提供揭示方法局限性的实例，并讨论了由结论导出的一般闭包性质。