EnormousInt.12pt.6_1_00

# EnormousInt.12pt.6_1_00 - 1 ENORMOUS INTEGERS IN REAL LIFE...

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1 ENORMOUS INTEGERS IN REAL LIFE by Harvey M. Friedman www.math.ohio-state.edu/~friedman/ June 1, 2000 1.F(x 1 ,...,x k ) = F(x 2 ,...,x k+1 ) N = the nonnegative integers. THEOREM 1.Let F:N k {1,. ..,r}. There exists x 1 < . .. < x k+1 such that F(x 1 ,...,x k ) = F(x 2 ,...,x k+1 ). This is an immediate conse-quence of a more general combinatorial theorem called Ramsey’s theorem, but it is much simpler to state. We call this adjacent Ramsey theory. There are inherent finite estimates here. THEOREM 1.2. For all k,r there exists t such that the following holds. Let F:N k {1,. ..,r}. There exists x 1 < . .. < x k+1 £ t such that F(x 1 ,...,x k ) = F(x 2 ,...,x k+1 ). QUESTION: What is the least such t = Adj(k,r)? THEOREM 1.3. Adj(k,1) = 1. Adj(k,2) = 2k+1. THEOREM 1.4. Let k 5. Adj(k,3) is greater than an exponential stack of k-2 1.5’s topped off with k-1. E.g., Adj(6,3) > 10 173 , Adj(7,3) > 10^10 172 . THEOREM 1.5. Adj(k,r) is at most an exponential stack of k-1 2’s topped off with a rea-sonable function of k and r. Laziness prevented me from being more precise than this. The related literature - upper bounds for higher Ramsey numbers - is virtually all asymptotic, so I can’t just quote it. Our adjacent Ramsey theory from the 80’s is lurking in the background in “Shift graphs and lower bounds on Ramsey numbers r k (l;r),” Duffus, Lefmann, Rodl, Discrete Mathematics 137 (1995), 177- 187.

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2 2. THE ACKERMAN HIERARCHY There is a good notation for really big numbers - up to a point. We use a streamlined version of the Ackerman hierarchy. Let f:Z + Z + be strictly increasing. We define the critical function f’:Z + Z + of f by: f’(n) = the result of applying f n times at 1. For n ≥ 1, the n-th function of the Ackerman hierarchy is the result of applying the ’ operator n-1 times starting at the doubling function. Thus f 1 is doubling, f 2 is exponentiation, f 3 is iterated exponentiation; i.e., f 3 (n) = E*(n) = an exponential stack of n 2’s. f 4 is confusing. We can equivalently present this by the recursion equations f 1 (n) = 2n, f k+1 (1) = f k (1), f k+1 (n+1) = f k (f k+1 (n)), where k,n ≥ 1. We define A(k,n) = f k (n). Note that A(k,1) = 2, A(k,2) = 4. For k 3, A(k,3) > A(k- 2,k-2), and as a function of k, eventually strictly dominates each f n , n 1. A(3,5) = 2 65,536 . A(4,3) = 65,536. A(4,4) = E*(65,536). And A(4,5) is E*(E*(65,536)). It seems safe to assert, e.g., that A(5,5) is incomprehen- sibly large. We propose this number as a sort of benchmark. 3. BOLZANO WEIERSTRASS We start with the usual statement of BW. THEOREM 3.1. Let x[1],x[2],. .. be an infinite sequence from the closed unit interval [0,1]. There exists k 1 < k 2 < . .. such that the subsequence x[k 1 ],x[k 2 ],. .. converges. We can obviously move towards estimations like this: THEOREM 3.2. Let x[1],x[2],. .. be an infinite sequence from the closed unit interval [0,1]. There exists k 1 < k 2 < . .. such that |x[k i+1 ]-x[k i ]| < 1/i 2 , i 1. But now we shake things up:
3 THEOREM 3.3. Let x[1],x[2],. .. be an infinite sequence from the closed unit interval [0,1]. There exists k 1 < k 2 < . ..

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EnormousInt.12pt.6_1_00 - 1 ENORMOUS INTEGERS IN REAL LIFE...

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