Distinct Volume Subsets: The Uncountable Case
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1 Distinct Volume Subsets: The Uncountable Case Douglas Ulrich 1 William Gasarch 2 1 University of California, Irvine ds ulrich@hotmail.com 2 University of Maryland at College Park gasarch@cs.umd.edu July 20, 2018 D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
2 Paul Erdös. Some remarks on set theory. Proceedings of the American Mathematical Society, 1: , D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
3 Paul Erdös. Some remarks on set theory. Proceedings of the American Mathematical Society, 1: , David Conlon, Jacob Fox, William Gasarch, David G. Harris, Douglas Ulrich, and Samuel Zbarsky. Distinct Volume Subsets, SIAM J. Discrete Math. 29(1), D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
4 Paul Erdös. Some remarks on set theory. Proceedings of the American Mathematical Society, 1: , David Conlon, Jacob Fox, William Gasarch, David G. Harris, Douglas Ulrich, and Samuel Zbarsky. Distinct Volume Subsets, SIAM J. Discrete Math. 29(1), Douglas Ulrich and William Gasarch. Distinct Volume Subsets via Indiscernibles, Archive for Mathematical Logic (to appear). D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
5 A Theorem of Erdös D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
6 A Theorem of Erdös Theorem (Erdös) Suppose X R d is infinite. Then there is Y X with Y = X, such that all pairs of points from Y have distinct distances. D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
7 A Theorem of Erdös Theorem (Erdös) Suppose X R d is infinite. Then there is Y X with Y = X, such that all pairs of points from Y have distinct distances. Proof when X is countable. D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
8 A Theorem of Erdös Theorem (Erdös) Suppose X R d is infinite. Then there is Y X with Y = X, such that all pairs of points from Y have distinct distances. Proof when X is countable. 1 Enumerate X = {a n : n < ω}. Define f : ( N 2) R via f ({n, m}) = a n a m. D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
9 A Theorem of Erdös Theorem (Erdös) Suppose X R d is infinite. Then there is Y X with Y = X, such that all pairs of points from Y have distinct distances. Proof when X is countable. 1 Enumerate X = {a n : n < ω}. Define f : ( N 2) R via f ({n, m}) = a n a m. 2 Apply the canonical Ramsey theorem to f to get some infinite Y X on which f is either homogeneous, or min-homogeneous, or max-homogeneous, or rainbow (injective). D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
10 A Theorem of Erdös Theorem (Erdös) Suppose X R d is infinite. Then there is Y X with Y = X, such that all pairs of points from Y have distinct distances. Proof when X is countable. 1 Enumerate X = {a n : n < ω}. Define f : ( N 2) R via f ({n, m}) = a n a m. 2 Apply the canonical Ramsey theorem to f to get some infinite Y X on which f is either homogeneous, or min-homogeneous, or max-homogeneous, or rainbow (injective). 3 If we get a rainbow set, then we are done. Otherwise, get a contradiction to Hilbert s basis theorem. D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
11 A Theorem of Erdös Theorem (Erdös) Suppose X R d is infinite. Then there is Y X with Y = X, such that all pairs of points from Y have distinct distances. Proof when X is countable. 1 Enumerate X = {a n : n < ω}. Define f : ( N 2) R via f ({n, m}) = a n a m. 2 Apply the canonical Ramsey theorem to f to get some infinite Y X on which f is either homogeneous, or min-homogeneous, or max-homogeneous, or rainbow (injective). 3 If we get a rainbow set, then we are done. Otherwise, get a contradiction to Hilbert s basis theorem. For example, if f is min-homogeneous, look at V n := {x C d : i<d (x(i) a m (i)) 2 = i<d (a n (i) a m (i)) 2 for all m < n}. D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
12 Cardinals D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
13 Cardinals We work in ZFC. D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
14 Cardinals We work in ZFC. The cardinal numbers are well-ordered: 0, 1, 2,..., D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
15 Cardinals We work in ZFC. The cardinal numbers are well-ordered: 0, 1, 2,..., ℵ 0, ℵ 1, ℵ 2,..., D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
16 Cardinals We work in ZFC. The cardinal numbers are well-ordered: 0, 1, 2,..., ℵ 0, ℵ 1, ℵ 2,..., ℵ ω, ℵ ω+1, ℵ ω+2,..., D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
17 Cardinals We work in ZFC. The cardinal numbers are well-ordered: 0, 1, 2,..., ℵ 0, ℵ 1, ℵ 2,..., ℵ ω, ℵ ω+1, ℵ ω+2,..., ℵ ω1, ℵ ω1 +1,... D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
18 Cardinals We work in ZFC. The cardinal numbers are well-ordered: 0, 1, 2,..., ℵ 0, ℵ 1, ℵ 2,..., ℵ ω, ℵ ω+1, ℵ ω+2,..., ℵ ω1, ℵ ω1 +1,..., Definition Suppose κ is an infinite cardinal. Let X be a set of cardinality κ. D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
19 Cardinals We work in ZFC. The cardinal numbers are well-ordered: 0, 1, 2,..., ℵ 0, ℵ 1, ℵ 2,..., ℵ ω, ℵ ω+1, ℵ ω+2,..., ℵ ω1, ℵ ω1 +1,..., Definition Suppose κ is an infinite cardinal. Let X be a set of cardinality κ. κ is singular if X can be split into fewer than κ pieces, each of size less than κ. D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
20 Cardinals We work in ZFC. The cardinal numbers are well-ordered: 0, 1, 2,..., ℵ 0, ℵ 1, ℵ 2,..., ℵ ω, ℵ ω+1, ℵ ω+2,..., ℵ ω1, ℵ ω1 +1,..., Definition Suppose κ is an infinite cardinal. Let X be a set of cardinality κ. κ is singular if X can be split into fewer than κ pieces, each of size less than κ. Otherwise, κ is regular. D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
21 Cardinals We work in ZFC. The cardinal numbers are well-ordered: 0, 1, 2,..., ℵ 0, ℵ 1, ℵ 2,..., ℵ ω, ℵ ω+1, ℵ ω+2,..., ℵ ω1, ℵ ω1 +1,..., Definition Suppose κ is an infinite cardinal. Let X be a set of cardinality κ. κ is singular if X can be split into fewer than κ pieces, each of size less than κ. Otherwise, κ is regular. 2 ℵ 0, the cardinality of the continuum, fits somewhere among the ℵ s. D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
22 Higher arity What Erdös Proved D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
23 Higher arity What Erdös Proved Definition Suppose X R d and 2 a d + 1. Then say that X is a-rainbow if for all u, v ( X a), if u and v both have non-zero volume, then they have distinct volumes. D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
24 Higher arity What Erdös Proved Definition Suppose X R d and 2 a d + 1. Then say that X is a-rainbow if for all u, v ( X a), if u and v both have non-zero volume, then they have distinct volumes. Theorem (Erdös) Suppose X R d is infinite, and X is regular, and 2 a d + 1. Then there is Y X with Y = X, such that Y is a-rainbow. D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
25 Higher arity What Erdös Proved Definition Suppose X R d and 2 a d + 1. Then say that X is a-rainbow if for all u, v ( X a), if u and v both have non-zero volume, then they have distinct volumes. Theorem (Erdös) Suppose X R d is infinite, and X is regular, and 2 a d + 1. Then there is Y X with Y = X, such that Y is a-rainbow. Theorem (Erdös) Suppose κ 2 ℵ 0 is singular. Then there is X R d of size κ, such that whenever Y X has size κ, then Y is not 3-rainbow. D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
26 Gameplan D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
27 Gameplan We want to control degenerate simplices, and we want to understand what happens at singular cardinals. D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
28 Gameplan We want to control degenerate simplices, and we want to understand what happens at singular cardinals. Given X R d, choose a finite coloring f : X 2d+2 c such that f (w) encodes the set of all subsets of w of zero volume, and the set of all pairs of subsets of w of equal volume. D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
29 Gameplan We want to control degenerate simplices, and we want to understand what happens at singular cardinals. Given X R d, choose a finite coloring f : X 2d+2 c such that f (w) encodes the set of all subsets of w of zero volume, and the set of all pairs of subsets of w of equal volume. Our goal is to find some Y X such that f behaves canonically on Y. D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
30 Gameplan We want to control degenerate simplices, and we want to understand what happens at singular cardinals. Given X R d, choose a finite coloring f : X 2d+2 c such that f (w) encodes the set of all subsets of w of zero volume, and the set of all pairs of subsets of w of equal volume. Our goal is to find some Y X such that f behaves canonically on Y. If f were some arbitrary coloring, we would need to first fix an ordering of X. We would then apply the Erdös-Rado theorem to get some Y X such that f is constant on increasing tuples from Y ; typically Y << X. D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
31 Gameplan We want to control degenerate simplices, and we want to understand what happens at singular cardinals. Given X R d, choose a finite coloring f : X 2d+2 c such that f (w) encodes the set of all subsets of w of zero volume, and the set of all pairs of subsets of w of equal volume. Our goal is to find some Y X such that f behaves canonically on Y. If f were some arbitrary coloring, we would need to first fix an ordering of X. We would then apply the Erdös-Rado theorem to get some Y X such that f is constant on increasing tuples from Y ; typically Y << X. But note that our coloring f is definable in (C, +,, 0, 1), a stable structure. This gives us added leverage. D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
32 Stability Definition Suppose M is a structure. Then M is stable if: for all N M, for all d < ω, and for all definable X N d N d, X does not contain an infinite half-graph. D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
33 Stability Definition Suppose M is a structure. Then M is stable if: for all N M, for all d < ω, and for all definable X N d N d, X does not contain an infinite half-graph. In more words, there is no sequence (a n : n < ω) from N d such that for all n, m < ω, (a n, a m ) X if and only if n < m. D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
34 Stability Definition Suppose M is a structure. Then M is stable if: for all N M, for all d < ω, and for all definable X N d N d, X does not contain an infinite half-graph. In more words, there is no sequence (a n : n < ω) from N d such that for all n, m < ω, (a n, a m ) X if and only if n < m. Example (C, +,, 0, 1) is a stable structure. D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
35 Stability Definition Suppose M is a structure. Then M is stable if: for all N M, for all d < ω, and for all definable X N d N d, X does not contain an infinite half-graph. In more words, there is no sequence (a n : n < ω) from N d such that for all n, m < ω, (a n, a m ) X if and only if n < m. Example (C, +,, 0, 1) is a stable structure. Note: the volume of a simplex in R d is the square root of a certain polynomial (the Cayley-Menger determinant) in the coefficients of the vertices. Thus the coloring f on the previous slide really is definable in (C, +,, 0, 1). D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
36 Canonization for regular κ D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
37 Canonization for regular κ Theorem (Shelah) Suppose M is a stable structure, X M has regular cardinality, and f : M n c is a definable coloring. Then there is Y X with Y = X such that f is constant on tuples of distinct elements from Y. D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
38 Canonization for regular κ Theorem (Shelah) Suppose M is a stable structure, X M has regular cardinality, and f : M n c is a definable coloring. Then there is Y X with Y = X such that f is constant on tuples of distinct elements from Y. Definition Suppose X R d and 2 a d + 1. Then say that X is strongly a-rainbow if for all distinct u, v ( X a), the volumes of u and v are distinct and nonzero. D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
39 Canonization for regular κ Theorem (Shelah) Suppose M is a stable structure, X M has regular cardinality, and f : M n c is a definable coloring. Then there is Y X with Y = X such that f is constant on tuples of distinct elements from Y. Definition Suppose X R d and 2 a d + 1. Then say that X is strongly a-rainbow if for all distinct u, v ( X a), the volumes of u and v are distinct and nonzero. Say that X is strictly a-rainbow if X is strongly a -rainbow for all a a, and the volume of every a + 1-element subset of X is 0 (i.e., X is a subset of an a 1-dimensional hyperplane). D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
40 Canonization for regular κ Theorem (Shelah) Suppose M is a stable structure, X M has regular cardinality, and f : M n c is a definable coloring. Then there is Y X with Y = X such that f is constant on tuples of distinct elements from Y. Definition Suppose X R d and 2 a d + 1. Then say that X is strongly a-rainbow if for all distinct u, v ( X a), the volumes of u and v are distinct and nonzero. Say that X is strictly a-rainbow if X is strongly a -rainbow for all a a, and the volume of every a + 1-element subset of X is 0 (i.e., X is a subset of an a 1-dimensional hyperplane). Theorem (U., G.) Suppose X R d is infinite, with X regular. Then there is Y X with Y = X, such that for some 2 a d + 1, Y is strictly a-rainbow. D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
41 Singular κ D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
42 Singular κ Theorem (U., G.) Suppose M is a stable structure, and X M has singular cardinality κ, and f : M n c is definable. D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
43 Singular κ Theorem (U., G.) Suppose M is a stable structure, and X M has singular cardinality κ, and f : M n c is definable. Let E be any equivalence relation on X with fewer than κ classes, each of size less than κ. D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
44 Singular κ Theorem (U., G.) Suppose M is a stable structure, and X M has singular cardinality κ, and f : M n c is definable. Let E be any equivalence relation on X with fewer than κ classes, each of size less than κ. Then there is Y X with Y = X, such that for all a X n, f (a) depends only on the isomorphism type of (range(a), E). D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
45 Singular κ Theorem (U., G.) Suppose M is a stable structure, and X M has singular cardinality κ, and f : M n c is definable. Let E be any equivalence relation on X with fewer than κ classes, each of size less than κ. Then there is Y X with Y = X, such that for all a X n, f (a) depends only on the isomorphism type of (range(a), E). Theorem (U., G.) Suppose κ 2 ℵ 0 is infinite. Then there is a finite list (C i : i < i ) of subsets of R d of size κ, such that whenever X R d has size κ, there is some i < i and some injection F : C i X, which preserves the relations u is a degenerate simplex and u, v have the same volume, for all tuples u, v of length 2 a d + 1. D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
46 Definable subsets of R d D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
47 Definable subsets of R d Definition Suppose X is a topological space. Then P X is perfect is P is closed and has no nonisolated points. D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
48 Definable subsets of R d Definition Suppose X is a topological space. Then P X is perfect is P is closed and has no nonisolated points. Any reasonably definable subset of R d (for instance, any Borel subset) is either countable or else has a perfect subset. D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
49 Definable subsets of R d Definition Suppose X is a topological space. Then P X is perfect is P is closed and has no nonisolated points. Any reasonably definable subset of R d (for instance, any Borel subset) is either countable or else has a perfect subset. Theorem (Blass) Suppose f : ( ) ( 2 N n c partitions 2 N) n into finitely many pieces, each with the property of Baire. Then there is some perfect P 2 N on which f takes on one of (n 1)! canonical behaviors. D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
50 Definable subsets of R d Definition Suppose X is a topological space. Then P X is perfect is P is closed and has no nonisolated points. Any reasonably definable subset of R d (for instance, any Borel subset) is either countable or else has a perfect subset. Theorem (Blass) Suppose f : ( ) ( 2 N n c partitions 2 N) n into finitely many pieces, each with the property of Baire. Then there is some perfect P 2 N on which f takes on one of (n 1)! canonical behaviors. Theorem (U., G.) Suppose P R d is perfect. Then there is Q P perfect and some 2 a d + 1, such that Q is strictly a-rainbow. D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
51 A Relative Consistency Result D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
52 A Relative Consistency Result By Erdös s theorem and our perfect set theorem, the following holds in ZFC as well as ZF + antichoice principles: D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
53 A Relative Consistency Result By Erdös s theorem and our perfect set theorem, the following holds in ZFC as well as ZF + antichoice principles: Theorem Given X R d infinite, there is Y X with Y = X, such that Y is 2-rainbow. D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
54 A Relative Consistency Result By Erdös s theorem and our perfect set theorem, the following holds in ZFC as well as ZF + antichoice principles: Theorem Given X R d infinite, there is Y X with Y = X, such that Y is 2-rainbow. Nonetheless D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
55 A Relative Consistency Result By Erdös s theorem and our perfect set theorem, the following holds in ZFC as well as ZF + antichoice principles: Theorem Given X R d infinite, there is Y X with Y = X, such that Y is 2-rainbow. Nonetheless Theorem (U., G.) It is consistent with ZF that there is some uncountable X R d which has no uncountable 2-rainbow subset. D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
56 Thank you D. Ulrich and W. Gasarch (UMCP and UCI) Distinct Volume Subsets July 20, / 12
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