[FOM] 387: Better Polynomial Shift Translation/typos

[Harvey Friedman]

We have found a better Polynomial Shift Translation Theorem by simply  
replacing "positive translation" with "translation". It was idiotic  
that I didn't see this earlier.

BUT first let me correct two typos in http://www.cs.nyu.edu/pipermail/fom/2010-January/014282.html

I forgot to erase two occurrences of "If". Here is the correction.

UPPER SHIFT GREEDY CLIQUE SEQUENCES

Let G be a simple graph on Q^k. An Upper Shift Greedy Clique Sequence
for G is a nonempty sequence x[1],...,x[n] from Q^k with the following
properties.

i. x[1] is the 0 tuple in Q^k.
ii. Let 2 <= 2m <= n-1. Let y be the m-th subsequence of length k of
the concatenation of x[1],...,x[2m-1]. Then
    a. x[2m] <= y and (y,x[2m]) is not an edge of G.
    b. x[2m+1] is the upper shift of x[2m].
iii. {x[2],...,x[n]} is a clique in G.

UPPER SHIFT GREEDY DOWN CLIQUE SEQUENCES

Let G be a digraph on Q^k. An Upper Shift Greedy Down Clique Sequence
for G is a nonempty sequence x[1],...,x[n] from Q^k with the following
properties.

i. x[1] is the 0 vector in Q^k.
ii. Let 2 <= 2m <= n-1. Let y be the m-th subsequence of length k of
the concatenation of x[1],...,x[2m-1]. Then
    a. x[2m] = y, or (x[2m] < y and (y,x[2m]) is not an edge of G).
    b. x[2m+1] is the upper shift of x[2m].
iii. {x[2],...,x[n]} is a down clique in G.

EXTREME UPPER SHIFT GREEDY DOWN CLIQUE SEQUENCES

***no change***

ALSO, the word "open" is quite stupid. Replace "open" with "terminal".  
I.e., terminal thread.

NOW for the restatement of the polynomials.

Let n_1,...,n_k be in Z and 1 <= i <= k. The translates of  
(n_1,...,n_k) in the i-th coordinate are obtained by adding an integer  
to the i-th coordinate.

POLYNOMIAL SHIFT TRANSLATION THEOREM. For all polynomials P:Z^k into
Z^k, there exist distinct positive integers n_1,...,n_k+1 such that,  
in each coordinate, the number of translates of (n_1,...,n_k) achieved  
by P is at most the number of translates of (n_2,...,n_k+1) achieved  
by P.

QUADRATIC SHIFT TRANSLATION THEOREM. For all quadratics P:Z^k into
Z^k, there exist distinct positive integers n_1,...,n_k+1 such that,  
in each coordinate, the number of translates of (n_1,...,n_k) achieved  
by P is at most the number of translates of (n_2,...,n_k+1) achieved  
by P.

THEOREM. Both of the above theorems are provable in ACA' but not
in Peano Arithmetic. They each imply 2-Con(PA) over EFA.

Here ACA' is ACA_0 + "for all x,n, the n-th jump of x exists".

I think that they are equivalent to 3-Con(PA) over EFA.

**********************

I use http://www.math.ohio-state.edu/~friedman/ for downloadable
manuscripts. This is the 387th in a series of self contained numbered
postings to FOM covering a wide range of topics in f.o.m. The list of
previous numbered postings #1-349 can be found at http://www.cs.nyu.edu/pipermail/fom/2009-August/014004.html
in the FOM archives.

350: one dimensional set series  7/23/09  12:11AM
351: Mapping Theorems/Mahlo/Subtle  8/6/09  10:59PM
352: Mapping Theorems/simpler  8/7/09  10:06PM
353: Function Generation 1  8/9/09  12:09PM
354: Mahlo Cardinals in HIGH SCHOOL 1  8/9/09  6:37PM
355: Mahlo Cardinals in HIGH SCHOOL 2  8/10/09  6:18PM
356: Simplified HIGH SCHOOL and Mapping Theorem  8/14/09  9:31AM
357: HIGH SCHOOL Games/Update  8/20/09  10:42AM
358: clearer statements of HIGH SCHOOL Games  8/23/09  2:42AM
359: finite two person HIGH SCHOOL games  8/24/09  1:28PM
360: Finite Linear/Limited Memory Games  8/31/09  5:43PM
361: Finite Promise Games  9/2/09  7:04AM
362: Simplest Order Invariant Game  9/7/09  11:08AM
363: Greedy Function Games/Largest Cardinals 1
364: Anticipation Function Games/Largest Cardinals/Simplified 9/7/09
11:18AM
365: Free Reductions and Large Cardinals 1  9/24/09  1:06PM
366: Free Reductions and Large Cardinals/polished  9/28/09  2:19PM
367: Upper Shift Fixed Points and Large Cardinals  10/4/09  2:44PM
368: Upper Shift Fixed Point and Large Cardinals/correction  10/6/09
8:15PM
369. Fixed Points and Large Cardinals/restatement  10/29/09  2:23PM
370: Upper Shift Fixed Points, Sequences, Games, and Large Cardinals
11/19/09  12:14PM
371: Vector Reduction and Large Cardinals  11/21/09  1:34AM
372: Maximal Lower Chains, Vector Reduction, and Large Cardinals
11/26/09  5:05AM
373: Upper Shifts, Greedy Chains, Vector Reduction, and Large
Cardinals  12/7/09  9:17AM
374: Upper Shift Greedy Chain Games  12/12/09  5:56AM
375: Upper Shift Clique Games and Large Cardinals 1
376: The Upper Shift Greedy Clique Theorem, and Large Cardinals
12/24/09  2:23PM
377: The Polynomial Shift Theorem  12/25/09  2:39PM
378: Upper Shift Clique Sequences and Large Cardinals  12/25/09  2:41PM
379: Greedy Sets and Huge Cardinals 1
380: More Polynomial Shift Theorems  12/28/09  7:06AM
381: Trigonometric Shift Theorem  12/29/09  11:25AM
382: Upper Shift Greedy Cliques and Large Cardinals  12/30/09  2:51AM
383: Upper Shift Greedy Clique Sequences and Large Cardinals 1
12/30/09  3:25PM
384: THe Polynomial Shift Translation Theorem/CORRECTION  12/31/09
7:51PM
385: Shifts and Extreme Greedy Clique Sequences  1/1/10  7:35PM
386: Terrifically and Extremely Long Finite Sequences  1/1/10  7:35PM

Harvey Friedman