## Continuous Ramsey theory on Polish spaces and covering the plane by functions

### BibTeX

@MISC{Geschke_continuousramsey,

author = {Stefan Geschke and Martin Goldstern and Menachem Kojman},

title = {Continuous Ramsey theory on Polish spaces and covering the plane by functions},

year = {}

}

### OpenURL

### Abstract

Abstract. We investigate the Ramsey theory of continuous graph-structures on complete, separable metric spaces and apply the results to the problem of covering a plane by functions. Let the homogeneity number hm(c) of a pair-coloring c: [X] 2 → 2 be the number of c-homogeneous subsets of X needed to cover X. We isolate two continuous pair-colorings on the Cantor space 2 ω, cmin and cmax, which satisfy hm(cmin) ≤ hm(cmax) and prove: Theorem. (1) For every Polish space X and every continuous pair-coloring c: [X] 2 → 2 with hm(c)> ℵ0, hm(c) = hm(cmin) or hm(c) = hm(cmax). (2) There is a model of set theory in which hm(cmin) = ℵ1 and hm(cmax) =

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Citation Context ... . # 6. Concluding remarks and open problems The numbers hm(c min ), hm(cmax ), Cov(Cont(R)) and Cov(Lip(R)) are examples of covering numbers of meager ideals. Although the hope expressed by Blass in =-=[10] to find a-=- classification of all "simple" cardinal invariants of the continuum was shattered by the construction in [21] of uncountably many di#erent covering numbers of simply defined meager ideals, ... |

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Citation Context ...to prove that hm = 2 #0 if the continuum is a limit cardinal, but did not prove more about hm and eventually found a way to eliminate GCH which did not involve hm, which was consequently published in =-=[22]-=-. It is not clear why homogeneity numbers of continuous pair-colorings on Polish spaces were not studied earlier. We can only speculate about that. In the very short time since their study was begun, ... |

12 |
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Citation Context ...eventually dominates all functions from D := h −1 (x), by elementarity, there is no function at all which eventually dominates every function in D. In other words, D is unbounded. A result of Kechris =-=[20]-=- says that every unbounded and closed set D ⊆ ω ω satisfies D = A∪P, A∩P = ∅ where A is bounded, i.e., a single function eventually dominates all functions in A, and P is superperfect, i.e., for all s... |

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Applications of PCF theory
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Citation Context ...the relation to planar convex geometry and to finite random graphs, which were mentioned above, there are relations to large cardinals, determinacy and pcf theory. Quite recently, Shelah and Zapletal =-=[30]-=- defined n-dimensional generalizations of hm(c min ) and integrated forcing, pcf theory and determinacy theory to prove several duality theorem for those numbers. We do not know at the moment if # 1sh... |

8 |
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Citation Context ...le column. We draw attention to the fact that the rows (2)--(6) have to share at most two consecutive cardinals since Cov(Cont(2 # )) cannot be 1 This holds for every open coloring on an analytic set =-=[18]-=-. CONTINUOUS RAMSEY THEORY 3 (6) # Cov(Cont(2 # )) # + (5) 2 #0 (4) hm(cmax ) (3) Cov(Lip(R)) # Cov(Lip(# # )) = Cov(Lip(2 # )) = hm(c min ) (2) Cov(Cont(R)) = Cov(Cont(# # )) = Cov(Cont(2 # )) (1) d ... |

7 |
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Citation Context ...ependent set of size # # n#. This stands in strong contrast to a randomly chosen graph: in a random graph on n vertices there is almost certainly no clique and no independent set of size 2 log n (see =-=[5]-=-). Fact 2.24 (N. Alon). Every finite (induced) subgraph H of (# # , c parity ) satisfies that the chromatic number of H is equal to the maximal size of a clique in H. Proof. Two proofs of this fact ar... |

7 |
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Citation Context ...urally related to the broader class of open colorings, that has been of interest for set theorists for three decades now, and have motivated several important developments in the technique of forcing =-=[7, 8, 2]-=-. Open coloring axioms, which are statements in the Ramsey theory of open colorings, are frequently used set-theoretic axioms in the theory of the continuum (see [33, 32, 17, 27] and the references th... |

7 |
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Citation Context ...ogy. If X is a compact metric space, then so is Comp(X). Recall that compact metric spaces are Polish. The components of Comp(X) are singletons. Since Comp(X) is compact, it is zero-dimensional. (See =-=[15]-=- for this. We assume compact spaces to be Hausdor#.) Lemma 2.12. Let X be compact and c : [X ] 2 # 2 continuous. Define a coloring c : [Comp(X)] 2 # 2 by c # comp(x), comp(y) # = c(x, y) for all x, y ... |

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Citation Context ...s of covering numbers of meager ideals. Although the hope expressed by Blass in [10] to find a classification of all "simple" cardinal invariants of the continuum was shattered by the constr=-=uction in [21] of uncountably many-=- di#erent covering numbers of simply defined meager ideals, there is still hope to find the "largest" nontrivial covering number of a meager ideal. By "nontrivial" it is meant that... |

5 |
Problems and results
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Citation Context ...e recent survey [11] and [12]). The forcing construction that separatesshm(c min ) from hm(cmax ) (Section 4 below) makes a crucial use of a Ramsey connection between perfect graphs and random graphs =-=[4]-=-. Within set theory, continuous pair-colorings are naturally related to the broader class of open colorings, that has been of interest for set theorists for three decades now, and have motivated sever... |

5 | More on convexity numbers of closed sets
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(Show Context)
Citation Context ...vered by countably many convex subsets (namely, its convex subsets generate a proper #-ideal), it has a closed subset on which the convex subsets of the whole set generate a meager ideal (see [24] or =-=[19]-=-). For some closed subsets of the plane, this meager ideal coincides with the homogeneity ideal of some continuous pair coloring [20]. Saharon Shelah remarked recently to the authors that he came clos... |

5 |
Decomposing euclidean space with a small number of smooth sets
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Citation Context ...ns The problem of covering a Euclidean space by smaller geometric objects is well investigated. Klee [25] proved that no separable Banach space can be covered by fewer than 2 #0 hyperplanes. Steprans =-=[31]-=- proved the consistency of covering R n+1 by fewer than continuum smooth manifolds of dimension n. We recall that a point (x, y) # X 2 is covered by a function f : X # X if f(x) = y or f(y) = x. By f ... |

4 |
Probabilistic methods in combinatorics, Probability and
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Citation Context ...ing c max . We shall now define a maximal almost node-coloring. Recall that the random graph on # is, up to isomorphism, the only homogeneous and universal graph in the class of all graphs on #. (See =-=[16]-=- for some information on the random graph.) Universality means: every graph (#, E) is embeddable as an induced subgraph into the random graph (in particular, every finite graph is embeddable as an ind... |

4 |
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Citation Context ... in the technique of forcing [7, 8, 2]. Open coloring axioms, which are statements in the Ramsey theory of open colorings, are frequently used set-theoretic axioms in the theory of the continuum (see =-=[33, 32, 17, 27]-=- and the references therein). 1.1. The results. Two simple pair-colorings c min and c max are defined on the Cantor space, and are shown to satisfy for every Polish space X and every continuous c : [X... |

4 |
private communication
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(Show Context)
Citation Context ...ty ideal of some continuous pair coloring [20]. Saharon Shelah remarked recently to the authors that he came close to discovering the properties of hm in his investigations of monadic theory of order =-=[29]-=-. In an attempt to remove GCH from the proof in the last section of [28] Shelah found a proof from the assumption hm = 2 #0 . He was able to prove that hm = 2 #0 if the continuum is a limit cardinal, ... |

4 |
All ℵ1-dense sets of reals can be isomorphic, Fund
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Citation Context ...lly related to the broader class of open colorings, that has been a focus of interest for set theorists for three decades now, and motivated several important developments in the technique of forcing =-=[7, 8, 2]-=-. Open coloring axioms, which are statements in the Ramsey theory of open colorings, are among the more frequently used set-theoretic axioms in the theory of the continuum (see [33, 32, 17, 27] and th... |

3 |
Covering R n+1 by graphs of n-ary functions and long linear orderings of Turing degrees
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(Show Context)
Citation Context ...the result by Ciesielski and Pawlikowski on covering R 2 by continuously di#erentiable functions. Hart asked whether Cov(Lip(2 # )) can be di#erent from Cov(Cont(2 # )). Recently, Abraham and Geschke =-=[1]-=- proved that it is consistent to cover R n+1 by # n-ary continuous functions with 2 #0 = # +n . Let us state the following folklore result that was brought to the authors' attention by Ireneusz Rec#la... |

3 |
Convexity conditions and intersections with smooth functions
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(Show Context)
Citation Context ...ntiable function f : R # R and an infinite perfect set P # R such that the derivative of f is constantly 0 on P and no function which is twice di#erentiable intersects f # P in infinitely many points =-=[3]-=-. Since the derivative of f is 0 on P , no inverse of a di#erentiable function intersects f # P in more than finitely many points. It follows that already the graph of f is not included in the union o... |

3 |
Definable automorphisms of P(!)=Fin
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(Show Context)
Citation Context |

3 |
Determinacy and Cardinal Invariants, preprint (Geschke) II. Mathematisches Institut, Freie Universität Berlin, Arnimallee 3, 14195
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(Show Context)
Citation Context ...(1)–(5) it is consistent that the value at the row is ℵ1 and at all rows above the value is ℵ2. The forcing for separating (2) from (3) is a new example of an optimal forcing in the sense of Zapletal =-=[31]-=- for increasing a cardinal invariant while leaving small everything that can be left small.CONTINUOUS RAMSEY THEORY 3 (6) ( Cov(Cont(2ω )+ )) (5) 2 ℵ0 (4) hm(cmax) (3) Cov(Lip(R)) ≥ Cov(Lip(ω ω )) = ... |

2 |
Chains and antichains in P
- Baumgartner
- 1980
(Show Context)
Citation Context ...urally related to the broader class of open colorings, that has been of interest for set theorists for three decades now, and have motivated several important developments in the technique of forcing =-=[7, 8, 2]-=-. Open coloring axioms, which are statements in the Ramsey theory of open colorings, are frequently used set-theoretic axioms in the theory of the continuum (see [33, 32, 17, 27] and the references th... |

2 |
Convex decompositions in the plane, meagre ideals, and colorings of the irrationals
- Geschke, Kojman, et al.
- 2002
(Show Context)
Citation Context ... , hm(c) = hm(c min ) or hm(c) = hm(cmax ). (2) There is a model of set theory in which hm(c min ) = #1 and hm(cmax ) = # 2 . The consistency of hm(c min ) = 2 # 0 and of hm(cmax )s2 # 0 follows from =-=[20]-=-. We prove that hm(c min ) is equal to the covering number of (2 # ) 2 by graphs of Lipschitz functions and their reflections on the diagonal. An iteration of an optimal forcing notion associated to c... |

2 | A small transitive family of continuous functions on the Cantor set, arXiv e-print math.GN/0004024
- Hart, Steeg
(Show Context)
Citation Context ...ed to cover X 2 and by Cov(Lip(X)) denote the analogous number for Lipschitz functions. Hart and van der Steeg showed the consistency of covering (2 # ) 2 by fewer than continuum continuous functions =-=[23]-=-, a result that actually follows from Steprans' CONTINUOUS RAMSEY THEORY 13 result mentioned above using some easy arguments from the present article. Ciesielski and Pawlikowski observed that Steprans... |

2 |
Convexity ranks in higher dimensions, Fund
- Kojman
(Show Context)
Citation Context ...s not covered by countably many convex subsets (namely, its convex subsets generate a proper #-ideal), it has a closed subset on which the convex subsets of the whole set generate a meager ideal (see =-=[24]-=- or [19]). For some closed subsets of the plane, this meager ideal coincides with the homogeneity ideal of some continuous pair coloring [20]. Saharon Shelah remarked recently to the authors that he c... |

2 |
Descriptive set theory and definable forcing, Memoirs of the American
- Zapletal
(Show Context)
Citation Context ...-(5) it is consistent that the value at the row is # 1 and at all rows above the value is # 2 . The forcing for separating (2) from (3) is a new example of an optimal forcing in the sense of Zapletal =-=[34]-=- for increasing a cardinal invariant while leaving small everything that can be left small. The inequality Cov(Lip(2 # )) # hm(c min ) and the consistency of hm(c)s2 #0 for every Polish space X and co... |

2 |
Problems and results in Extremal
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- 2003
(Show Context)
Citation Context ... (ω ω , cparity) satisfies that the chromatic number of H is equal to the maximal size of a clique in H.12 STEFAN GESCHKE, MARTIN GOLDSTERN, AND MENACHEM KOJMAN Proof. Two proofs of this fact are in =-=[3]-=-. The proof we include here was suggested to us by Stevo Todorčević. Define a partial order on ω ω by η1 ≤ η2 iff η1 = η2 or ∆(η1, η2) is odd and η1 precedes η2 in the lexicographic ordering on ω ω . ... |

1 |
Chains and antichains in P(ω
- Baumgartner
- 1980
(Show Context)
Citation Context ...lly related to the broader class of open colorings, that has been a focus of interest for set theorists for three decades now, and motivated several important developments in the technique of forcing =-=[7, 8, 2]-=-. Open coloring axioms, which are statements in the Ramsey theory of open colorings, are among the more frequently used set-theoretic axioms in the theory of the continuum (see [33, 32, 17, 27] and th... |

1 |
personal communication, May 2002
- Shelah
(Show Context)
Citation Context ...ty ideal of some continuous pair coloring [20]. Saharon Shelah remarked recently to the authors that he came close to discovering the properties of hm in his investigations of monadic theory of order =-=[29]-=-. In an attempt to remove GCH from the proof in the last section of [28] Shelah found a proof from the assumption hm = 2 ℵ0 . He was able to prove that hm = 2 ℵ0 if the continuum is a limit cardinal, ... |

1 |
Duality and the pcf theory, preprint
- Shelah, Zapletal
(Show Context)
Citation Context ...the relation to planar convex geometry and to finite random graphs, which were mentioned above, there are relations to large cardinals, determinacy and pcf theory. Quite recently, Shelah and Zapletal =-=[27]-=- defined n-dimensional generalizations of hm(cmin) and integrated forcing, pcf theory and determinacy theory to prove several duality theorem for those numbers. We do not know at the moment if ℵ1 < hm... |