[FOM] What is OCA?

[James Hirschorn]

I had heard of this very exciting result of Farah, but never got around to
even looking at the paper until now. 

On Wednesday 06 February 2008 18:34, Kreinovich, Vladik wrote:
> Open Coloring Axiom (OCA) is also explicitly stated in the same Faraj's
> paper on p. 10.
>
> > From John Baldwin
> >
> > Can you tell us what OCA is?  

Actually, one needs to consider the history to answer this question. 

The Open Coloring Axiom (OCA) was a major breakthrough, introduced in:

[ARS] U. Abraham, M. Rubin, S. Shelah, "On the consistency of some 
      partition theorems for continuous colorings, and the structure of
      aleph-one -dense real order types", APAL 29 (1985), 123--206.

For a topological space X, let up(X) denote the collection of all unordered 
pairs of members of X equipped with the quotient topology of the symmetry 
relation on X x X (i.e. (x,y)~(y,x)). In the above paper, an _open coloring_ 
of up(X) simply refers to a finite open cover of up(X). For an open coloring 
V={U_0,...,U_n} of up(X), a subset A of X is called _V-homogeneous_ if there 
exists a color i=0,...,n such that up(A) is a subset of U_i. An axiom called 
the "Open Coloring Axiom (OCA)" is defined, in [ARS], as:

(OCA)   Let X be a second countable space of cardinality aleph-1. Then for
        every open coloring V of up(X), X has a countable decomposition 
        into V-homogeneous pieces.

It is one of a number simple partition properties for second countable spaces 
introduced there. Despite its simplicity it is a powerful axiom which entails 
a number of consequences of the "heavy duty" forcing axiom PFA. For example, 
(letting MA denote Martin's Axiom) a remarkable result in [ARS] is:

Theorem A. MA + OCA implies that the continuum has cardinality aleph-2.

One of the other partition properties considered in [ARS] was called 
the "Semiopen coloring axiom (SOCA)". A partition 

   up(X) = K_0 U K_1 

of up(X) into two pieces is called _semiopen_ if K_0 is an open set. A subset 
A of X is called _i-homogeneous_ (wrt K_0 U K_1) if there is a color i=0,1 
such that up(A) is a subset of K_i. 

(SOCA)  Let X be a second countable space. Then every semiopen partition 
        of up(X) has an uncountable subset A of X that is i-homogeneous 
        for either i=0 or i=1. 

The analogue of Theorem A for SOCA is false, as is demonstrated in [ARS] by 
showing that e.g. MA + SOCA + 2^{aleph-0} = aleph-3 is consistent (of no 
particular significance to aleph-3). 

Let us denote the following axiom by (*):

(*)     Let X be a second countable space of cardinality aleph-1. Then
        every semiopen partition of up(X) has either an uncountable
        0-homogeneous subset of X, or a countable decomposition of X into
        1-homogeneous pieces. 

The following positive consequence of MA + SOCA is established in [ARS, 
Proposition 1.4]. 

Theorem B. MA + SOCA implies (*).

Now, a very slight strengthening of (*) was much later introduced in:

[T]  S. Todorcevic, "Partition problems in topology", Contemporary  
     mathematics, vol. 84, AMS, (1989).

In [T], this strengthening is called---the "Open Coloring Axiom (OCA)". I will 
distinguish it from the above axiom OCA using a prime:

(OCA')  Let X be a second countable space. Then 
        every semiopen partition of up(X) has either an uncountable
        0-homogeneous subset of X, or a countable decomposition of X into
        1-homogeneous pieces.

Thus, the only difference between OCA' and * is that the restriction "of 
cardinality aleph-1" on X has been removed. Since I presently do not have 
access to [T], I will not attempt comment on what else is and is not, written 
in [T].

The "OCA" used in Farah's paper is OCA' above. 

There seems to be significant omissions in the account of the history of OCA 
given in Farah's paper (http://arxiv.org/abs/0705.3085v3). On the first page 
the main result is stated thus:

Theorem 1. Todorcevic’s Open Coloring Axiom, OCA, implies that all
automorphisms of the Calkin algebra of a separable Hilbert space are inner.

Then, the reader is referred to section 2.3 for the statement of OCA, where it 
is written:

“[...] The following axiom was introduced by Todorcevic in [T].

OCA. ... Equivalent definition to OCA' above ...

[...] An another [sic] axiom called OCA in which both K_0 and K_1 were 
required to be open was considered by Abraham, Rubin and Shelah [no citation 
given]. This variant of OCA will not be used here.”

No other information on the history of OCA is provided in his paper.

James Hirschorn

> >
> > On Tue, 5 Feb 2008, Nik Weaver wrote:
> >> Farah's
> >>
> >> http://front.math.ucdavis.edu/0705.3085