179:
296:
265:
181:. Moreover, assuming OCA, Baire space contains few "gaps" between sets of sequences — more specifically, that the only possible gaps are
285:
Proceedings of the international congress of mathematicians (ICM 2010), Hyderabad, India, August 19–27, 2010. Vol. II: Invited lectures
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387:
367:
325:
25:
382:
151:
377:
372:
313:
282:
Moore, Justin Tatch (2011), "Logic and foundations the proper forcing axiom", in Bhatia, Rajendra (ed.),
29:
207:(1985), "On the consistency of some partition theorems for continuous colorings, and the structure of ℵ
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can be partitioned into a countable number of subsets such that any pair from the same subset is black.
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140:
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128:
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61:
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147:
132:
33:
283:
139:. The full OCA is consistent with (but independent of) ZFC, and follows from the
96:
93:
60:
is colored either black or white, with the set of white pairs being open in the
75:
has an uncountable subset such that any pair from this subset is white; or
92:, replaces the uncountability condition in the first case with being a
260:, Studies in Logic, vol. 34, London: College Publications,
124:
324:, Contemporary Mathematics, vol. 84, Providence, RI:
56:
is a subset of the reals, and each pair of elements of
160:
291:, Hackensack, NJ: World Scientific, pp. 3–29,
317:
173:
37:
68:. The open coloring axiom states that either:
8:
36:: two different versions were introduced by
41:
165:
159:
107:can be stated equivalently for arbitrary
24:) is an axiom about coloring edges of a
185:and analogous (κ,ω)-gaps where κ is an
7:
248:, notes on lectures by Matteo Viale
162:
38:Abraham, Rubin & Shelah (1985)
14:
203:Abraham, Uri; Rubin, Matatyahu;
320:Partition problems in topology
146:OCA implies that the smallest
1:
326:American Mathematical Society
225:10.1016/0168-0072(84)90024-1
174:{\displaystyle \aleph _{2}}
409:
239:Carotenuto, Gemma (2013),
211:-dense real order types",
115:Relation to other axioms
242:An introduction to OCA
175:
213:Ann. Pure Appl. Logic
176:
388:Independence results
368:Axioms of set theory
158:
141:proper forcing axiom
137:axiom of determinacy
32:are a subset of the
18:open coloring axiom
171:
103:. Both OCA and OCA
85:A weaker version,
314:Todorčević, Stevo
298:978-981-4324-30-4
267:978-1-84890-050-9
123:can be proved in
42:Todorčević (1989)
400:
354:
323:
309:
290:
278:
249:
247:
235:
180:
178:
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170:
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154:has cardinality
129:analytic subsets
109:separable spaces
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383:Infinite graphs
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336:
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299:
288:
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252:
245:
238:
210:
205:Shelah, Saharon
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199:
192:
189:not less than ω
187:initial ordinal
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156:
155:
135:, and from the
122:
117:
106:
90:
50:
12:
11:
5:
406:
404:
396:
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390:
385:
380:
378:Graph coloring
375:
370:
360:
359:
356:
355:
334:
310:
297:
279:
266:
254:Kunen, Kenneth
250:
236:
219:(2): 123–206,
208:
198:
195:
190:
183:Hausdorff gaps
168:
164:
120:
116:
113:
104:
88:
83:
82:
76:
62:complete graph
49:
46:
13:
10:
9:
6:
4:
3:
2:
405:
394:
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374:
373:Real analysis
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335:0-8218-5091-1
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148:unbounded set
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98:
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59:
55:
52:Suppose that
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45:
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31:
27:
23:
20:(abbreviated
19:
319:
284:
257:
241:
216:
212:
145:
133:Polish space
118:
100:
86:
84:
78:
72:
65:
57:
53:
51:
34:real numbers
21:
17:
15:
393:Determinacy
152:Baire space
97:perfect set
362:Categories
352:0659.54001
307:1258.03075
276:1262.03001
258:Set theory
233:0585.03019
197:References
163:ℵ
48:Statement
316:(1989),
256:(2011),
30:vertices
344:0980949
94:compact
40:and by
350:
342:
332:
305:
295:
274:
264:
231:
28:whose
289:(PDF)
246:(PDF)
131:of a
26:graph
330:ISBN
293:ISBN
262:ISBN
127:for
16:The
348:Zbl
303:Zbl
272:Zbl
229:Zbl
221:doi
150:of
125:ZFC
119:OCA
99:in
87:OCA
64:on
22:OCA
364::
346:,
340:MR
338:,
328:,
301:,
270:,
227:,
217:29
215:,
193:.
143:.
111:.
44:.
223::
209:1
191:2
167:2
121:P
105:P
101:X
89:P
79:X
73:X
66:X
58:X
54:X
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