We prove a general Ramsey theorem for trees with a successor operation. This theorem is a common generalization of the Carlson-Simpson Theorem and the Milliken Tree Theorem for regularly branching trees. Our theorem has a number of applications both in finite and infinite combinatorics. For example, we give a short proof of the unrestricted Ne\v{s}et\v{r}il-R\"odl theorem, and we recover the Graham-Rothschild theorem. Our original motivation came from the study of big Ramsey degrees - various trees used in the study can be viewed as trees with a successor operation. To illustrate this, we give a non-forcing proof of a theorem of Zucker on big Ramsey degrees.

翻译：我们证明了一个关于带有后继运算树的广义Ramsey定理。该定理是Carlson-Simpson定理和规则分支树的Milliken树定理的共同推广。我们的定理在有限和无限组合学中均有若干应用。例如，我们给出了无限制Nešetřil-Rödl定理的简短证明，并恢复了Graham-Rothschild定理。最初的研究动机源于大Ramsey度的研究——该研究中使用的各种树均可视为带有后继运算的树。为说明这一点，我们给出了Zucker关于大Ramsey度定理的一个非力迫证明。