[FOM] 506: Finite Embedded Weakly Maximal Cliques

Tue Oct 23 00:53:52 EDT 2012

THIS RESEARCH WAS PARTIALLY SUPPORTED BY THE JOHN TEMPLETON FOUNDATION

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THIS POSTING CONTINUES
http://www.cs.nyu.edu/pipermail/fom/2012-September/016702.html

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FINITE EMBEDDED WEAKLY MAXIMAL CLIQUES
by
Harvey M. Friedman
October 23, 2012

We have been focusing on

EVERY ORDER INVARIANT GRAPH ON Q>=0^k HAS AN f EMBEDDED MAXIMAL CLIQUE

for very simple partial f:Q into Q. See
http://www.cs.nyu.edu/pipermail/fom/2012-October/016742.html

The finite forms we have been working with involve obvious associated
algorithms for constructing f embedded maximal cliques. See
http://www.cs.nyu.edu/pipermail/fom/2012-September/016702.html

This approach to finite forms is here to stay because they set the
stage for the computer investigations discussed in
http://www.cs.nyu.edu/pipermail/fom/2012-September/016702.html

Nevertheless, we have continued to be interested in other kinds of
explicitly Pi01 independence results.

The kind of finite forms we present here take the following shape:

EVERY ORDER INVARIANT GRAPH ON Q>=0^k HAS A FINITE f EMBEDDED WEAKLY
MAXIMAL CLIQUE

where there is an obvious upper bound on the size of the clique. In
fact, we can place obvious upper bounds on the size of the numerators
and denominators, so that the statement becomes explicitly Pi01.

It remains to define "weakly maximal cliques".

Let G be an order invariant graph on Q>=0^k. For this, we use the
equivalence relation ~ for subsets of Q^n given by

S ~ S'i if and only if every element of S is order equivalent to an
element of S', and every element of S' is order equivalent to an
element of S.

We consider five natural maximality conditions on a clique S (in G).
For every clique S' containing S,

1. S' = S.
2. S' ~ S.
3. S'^k ~ S^k.
4. S' x {(0,...,k)} ~ S x {(0,...,k)}.
5. S'^k x {(0,...,k)} ~ S^k x {(0,...,k)}.

If we use 1, then we have ordinary maximality, and the statement is
refutable in EFA, because generally, S must be infinite.

If we use any of 2-4, then the statement is true, but we suspect is
provable already in ZFC.

Thus we focus on using 5, arriving at the notion "S is a weakly
maximal clique".

PROPOSITION 1. Let f be +1 on {0,...,n}, extended by the identity on Q
intersect (n+1,infinity). Every order invariant graph on Q>=0^k has a
finite f embedded weakly maximal clique.

PROPOSITION 2. Let partial order theoretic f:Q into Q be strictly
increasing and bicontinuous at all but at most one point. Every order
invariant graph on Q>=0^k has a finite f embedded weakly maximal
clique.

Recall these infinite forms.

PROPOSITION 3. Let f be +1 on {0,...,n}, extended by the identity on Q
intersect (n+1,infinity). Every order invariant graph on Q>=0^k has an
f embedded maximal clique.

PROPOSITION 4. Let partial order theoretic f:Q into Q be strictly
increasing and bicontinuous at all but at most one point. Every order
invariant graph on Q>=0^k has an f embedded maximal clique.

THEOREM 5. Propositions 1-4 are provably equivalent to Con(SRP) over
RCA_0. In the case of Propositions 1,2, we can use EFA instead of
RCA_0.

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I use http://www.math.ohio-state.edu/~friedman/ for downloadable
manuscripts. This is the 506th 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 #1-449 can be found
in the FOM archives at
http://www.cs.nyu.edu/pipermail/fom/2010-December/015186.html

450: Maximal Sets and Large Cardinals II  12/6/10  12:48PM
451: Rational Graphs and Large Cardinals I  12/18/10  10:56PM
452: Rational Graphs and Large Cardinals II  1/9/11  1:36AM
453: Rational Graphs and Large Cardinals III  1/20/11  2:33AM
454: Three Milestones in Incompleteness  2/7/11  12:05AM
455: The Quantifier "most"  2/22/11  4:47PM
456: The Quantifiers "majority/minority"  2/23/11  9:51AM
457: Maximal Cliques and Large Cardinals  5/3/11  3:40AM
458: Sequential Constructions for Large Cardinals  5/5/11  10:37AM
459: Greedy CLique Constructions in the Integers  5/8/11  1:18PM
460: Greedy Clique Constructions Simplified  5/8/11  7:39PM
461: Reflections on Vienna Meeting  5/12/11  10:41AM
462: Improvements/Pi01 Independence  5/14/11  11:53AM
463: Pi01 independence/comprehensive  5/21/11  11:31PM
464: Order Invariant Split Theorem  5/30/11  11:43AM
465: Patterns in Order Invariant Graphs  6/4/11  5:51PM
466: RETURN TO 463/Dominators  6/13/11  12:15AM
467: Comment on Minimal Dominators  6/14/11  11:58AM
468: Maximal Cliques/Incompleteness  7/26/11  4:11PM
469: Invariant Maximality/Incompleteness  11/13/11  11:47AM
470: Invariant Maximal Square Theorem  11/17/11  6:58PM
471: Shift Invariant Maximal Squares/Incompleteness  11/23/11  11:37PM
472. Shift Invariant Maximal Squares/Incompleteness  11/29/11  9:15PM
473: Invariant Maximal Powers/Incompleteness 1  12/7/11  5:13AMs
474: Invariant Maximal Squares  01/12/12  9:46AM
475: Invariant Functions and Incompleteness  1/16/12  5:57PM
476: Maximality, CHoice, and Incompleteness  1/23/12  11:52AM
477: TYPO  1/23/12  4:36PM
478: Maximality, Choice, and Incompleteness  2/2/12  5:45AM
479: Explicitly Pi01 Incompleteness  2/12/12  9:16AM
480: Order Equivalence and Incompleteness
481: Complementation and Incompleteness  2/15/12  8:40AM
482: Maximality, Choice, and Incompleteness 2  2/19/12 7:43AM
483: Invariance in Q[0,n]^k  2/19/12  7:34AM
484: Finite Choice and Incompleteness  2/20/12  6:37AM__
485: Large Large Cardinals  2/26/12  5:55AM
486: Naturalness Issues  3/14/12  2:07PM
487: Invariant Maximality/Naturalness  3/21/12  1:43AM
488: Invariant Maximality Program  3/24/12  12:28AM
489: Invariant Maximality Programs  3/24/12  2:31PM
490: Invariant Maximality Program 2  3/24/12  3:19PM
491: Formal Simplicity  3/25/12  11:50PM
492: Invariant Maximality/conjectures  3/31/12  7:31PM
493: Invariant Maximality/conjectures 2  3/31/12  7:32PM
494: Inv Max Templates/Z+up, upper Z+ equiv  4/5/12  4:17PM
495: Invariant Finite Choice  4/5/12  4:18PM
496: Invariant Finite Choice/restatement  4/8/12  2:18AM
497: Invariant Maximality Restated  5/2/12 2:49AM
498: Embedded Maximal Cliques 1  9/18/12  12:43AM
499. Embedded Maximal Cliques 2  9/19/12  2:50AM
500: Embedded Maximal Cliques 3  9/20/12  10:15PM
501: Embedded Maximal Cliques 4  9/23/12  2:16AM
502: Embedded Maximal Cliques 5  9/26/12  1:21AM
503: Proper Classes of Graphs  10/13/12  12:17PM
504. Embedded Maximal Cliques 6  10/14/12  12:49PM
505: Function Transfer Theory 10/21/12  2:15AM

Harvey Friedman