We investigate the uniform computational content of the open and clopen Ramsey theorems in the Weihrauch lattice. While they are known to be equivalent to $\mathrm {ATR_0}$ from the point of view of reverse mathematics, there is not a canonical way to phrase them as multivalued functions. We identify eight different multivalued functions (five corresponding to the open Ramsey theorem and three corresponding to the clopen Ramsey theorem) and study their degree from the point of view of Weihrauch, strong Weihrauch, and arithmetic Weihrauch reducibility. In particular one of our functions turns out to be strictly stronger than any previously studied multivalued functions arising from statements around $\mathrm {ATR}_0$ .

中文翻译：

---

Weihrauch 格中的开和闭 Ramsey 定理

我们研究了 Weihrauch 晶格中开和 cloopen Ramsey 定理的统一计算内容。虽然已知它们等同于$\mathrm {ATR_0}$从逆向数学的角度来看，没有一种规范的方法可以将它们表述为多值函数。我们确定了 8 个不同的多值函数（5 个对应于开 Ramsey 定理，3 个对应于 cloopen Ramsey 定理）并从 Weihrauch、强 Weihrauch 和算术 Weihrauch 可约性的角度研究它们的度数。特别是，我们的函数之一被证明比任何先前研究的多值函数都严格地强于由周围语句产生的$\mathrm {ATR}_0$.