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Does an existence of large cardinals have implications in more down-to-earth fields like number theory, finite combinatorics, graph theory, Ramsey theory or computability theory? Are there any ...

Let $(A_{n},*)$ denote the $n$-th classical Laver table. Let $X_{n}$ be the set of all finite sequences of elements from $A_{n}$. Define a function $E_{n}:X_{n}\rightarrow X_{n}$ by letting $E_{n}((...