Block subsequence theorem

n(4) is a large combinatorial number involved in Harvey Friedman's Block Subsequence Theorem.

Suppose we have a string x1, x2, ... made of an alphabet of k letters, such that no block of letters xi, ..., x2i is a subsequence of any later block xj, ..., x2j. Friedman showed that such a sequence cannot be infinite, and for every k there is a sequence of maximal length. Call this length n(k).


 * \(n(1) = 3\), using the string "111". "1111" does not satisfy the conditions since x1, x2 = "11" is a subsequence of x2, x3, x4 = "111".
 * \(n(2) = 11\), using the string "12221111111".
 * \(n(3)\) and \(n(4)\) are much larger. Let A be a version of the Ackermann function as follows:
 * \(A(1, n) = 2n\)
 * \(A(k + 1, n) = 2 ↑^k n\)
 * \(A(n) = A(n, n)\)
 * \(A(7198, 158386) \leq n(3) \leq A(A(5))\)
 * \(n(4) \geq A^{A(187196)}(1)\)