The open coloring axiom (abbreviated OCA) is an axiom about coloring edges of a
graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
whose
vertices are a subset of the
real numbers
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
: two different versions were introduced by and by .
Statement
Suppose that ''X'' is a subset of the reals, and each pair of elements of ''X'' is colored either black or white, with the set of white pairs being open in the
complete graph
In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices is c ...
on ''X''. The open coloring axiom states that either:
#''X'' has an uncountable subset such that any pair from this subset is white; or
#''X'' can be partitioned into a countable number of subsets such that any pair from the same subset is black.
A weaker version, OCA
P, replaces the uncountability condition in the first case with being a
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
perfect set
In general topology, a subset of a topological space is perfect if it is closed and has no isolated points. Equivalently: the set S is perfect if S=S', where S' denotes the set of all Limit point, limit points of S, also known as the derived se ...
in ''X''. Both OCA and OCA
P can be stated equivalently for arbitrary
separable space
In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence \_^ of elements of the space such that every nonempty open subset of the space contains at least one element of the ...
s.
Relation to other axioms
OCA
P can be proved in
ZFC for
analytic subsets of a
Polish space
In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named bec ...
, and from the
axiom of determinacy
In mathematics, the axiom of determinacy (abbreviated as AD) is a possible axiom for set theory introduced by Jan Mycielski and Hugo Steinhaus in 1962. It refers to certain two-person topological games of length ω. AD states that every game of ...
. The full OCA is consistent with (but independent of) ZFC, and follows from the
proper forcing axiom In the mathematical field of set theory, the proper forcing axiom (''PFA'') is a significant strengthening of Martin's axiom, where forcings with the countable chain condition (ccc) are replaced by proper forcings.
Statement
A forcing or parti ...
.
OCA implies that the smallest
unbounded set
:''"Bounded" and "boundary" are distinct concepts; for the latter see boundary (topology). A circle in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary.
In mathematical analysis and related areas of ma ...
of
Baire space
In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior.
According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are ...
has cardinality
. Moreover, assuming OCA, Baire space contains few "gaps" between sets of sequences — more specifically, that the only possible gaps are
Hausdorff gap
In mathematics, a Hausdorff gap consists roughly of two collections of sequences of integers, such that there is no sequence lying between the two collections. The first example was found by . The existence of Hausdorff gaps shows that the partia ...
s and analogous (κ,ω)-gaps where κ is an
initial ordinal
In a written or published work, an initial capital, also referred to as a drop capital or simply an initial cap, initial, initcapital, initcap or init or a drop cap or drop, is a letter at the beginning of a word, a chapter, or a paragraph that ...
not less than ω
2.
References
*
*
*
*
*{{citation , MR=0980949 , zbl=0659.54001 , last=Todorčević , first=Stevo , author-link=Stevo Todorčević , title=Partition problems in topology , series=Contemporary Mathematics , volume=84 , publisher=
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
, location=Providence, RI , year=1989 , isbn=0-8218-5091-1 , url-access=registration , url=https://archive.org/details/partitionproblem0000todo
Axioms of set theory
Real analysis
Graph coloring
Infinite graphs
Independence results
Determinacy