Slice - Revision history

89.138.38.66 at 20:47, 21 February 2009

2009-02-21T20:47:56Z

← Older revision		Revision as of 13:47, 21 February 2009
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	~~A '''slice''' of the cube~~ <math>~~[3]^~~n</math> ~~is a set of the form~~ <math>~~\Gamma_{~~a,b,c}</math> ~~for some (a,b,c)~~ in the triangular grid		For <math>n\in{\Bbb Z}_+</math> and <math>(a,b,c)</math> in the triangular grid

	:<math>\Delta_n := \{ (a,b,c) \in {\Bbb Z}_+^3: a+b+c = n \},</math>		:<math>\Delta_n := \{ (a,b,c) \in {\Bbb Z}_+^3: a+b+c = n \},</math>

	~~where~~ <math>\Gamma_{a,b,c}</math> is the set of all strings with a 1s, b 2s, and c ~~3s; thus~~, for instance,		the '''slice''' <math>\Gamma_{a,b,c}</math> is the set of all strings in <math>[3]^n</math> with <math>a</math> occurences of <math>1</math>, <math>b</math> occurences of <math>2</math>, and <math>c</math> occurences of <math>3</math>. Thus, for instance,

	:<math>\Gamma_{1,2,0} = \{ 011, 101, 101 \}.</math>		:<math>\Gamma_{1,2,0} = \{ 011, 101, 101 \}.</math>
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	The <math>k=2</math> analogue of a slice, sometimes called a ''layer'', plays a role in the proof of [[Sperner's theorem]].		The <math>k=2</math> analogue of a slice, sometimes called a ''layer'', plays a role in the proof of [[Sperner's theorem]].

	A [[combinatorial line]] must touch three slices <math>\Gamma_{a+r,b,c}, \Gamma_{a,b+r,c}, \Gamma_{a,b,c+r}</math> corresponding to an equilateral triangle in <math>\Delta_n</math>. Conversely, one might hope that any sufficiently "rich" subsets of three slices <math>\Gamma_{a+r,b,c}, \Gamma_{a,b+r,c}, \Gamma_{a,b,c+r}</math> in an equilateral triangle will have many combinatorial lines between them. However there appear to be quite a lot of obstructions to this hope, consider e.g. the subset of <math>\Gamma_{a+r,b,c}</math> of strings whose first digit is 1, and the subsets of <math>\Gamma_{a,b+r,c}, \Gamma_{a,b,c+r}</math> of strings whose first digit is 2.		A [[combinatorial line]] must touch three slices <math>\Gamma_{a+r,b,c}, \Gamma_{a,b+r,c}, \Gamma_{a,b,c+r}</math> corresponding to an equilateral triangle in <math>\Delta_n</math>. Conversely, one might hope that any sufficiently "rich" subsets of three slices <math>\Gamma_{a+r,b,c}, \Gamma_{a,b+r,c}, \Gamma_{a,b,c+r}</math> in an equilateral triangle will have many combinatorial lines between them. However there appear to be quite a lot of obstructions to this hope, consider e.g. the subset of <math>\Gamma_{a+r,b,c}</math> of strings whose first digit is <math>1</math>, and the subsets of <math>\Gamma_{a,b+r,c}, \Gamma_{a,b,c+r}</math> of strings whose first digit is <math>2</math>.

89.138.38.66 at 20:40, 21 February 2009

2009-02-21T20:40:06Z

← Older revision		Revision as of 13:40, 21 February 2009
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	:<math>\Delta_n := \{ (a,b,c) \in {\Bbb Z}_+^3: a+b+c = n \},</math>		:<math>\Delta_n := \{ (a,b,c) \in {\Bbb Z}_+^3: a+b+c = n \},</math>

	where <math>\Gamma_{a,b,c}</math> is the set of all strings with a 1s, b 2s, and c 3s~~. Thus~~ for instance		where <math>\Gamma_{a,b,c}</math> is the set of all strings with a 1s, b 2s, and c 3s; thus, for instance,

	:<math>\Gamma_{1,2,0} = \{ 011, 101, 101 \}.</math>		:<math>\Gamma_{1,2,0} = \{ 011, 101, 101 \}.</math>
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	The cube <math>[3]^n</math> has <math>\frac{(n+1)(n+2)}{2}</math> slices, and each slice <math>\Gamma_{a,b,c}</math> has cardinality <math>\frac{n!}{a!b!c!}</math>.		The cube <math>[3]^n</math> has <math>\frac{(n+1)(n+2)}{2}</math> slices, and each slice <math>\Gamma_{a,b,c}</math> has cardinality <math>\frac{n!}{a!b!c!}</math>.

	~~Each~~ slice <math>\Gamma_{a,b,c}</math> is incident to <math>2^a+2^b+2^c-3</math> [[line\|combinatorial lines]].		The slice <math>\Gamma_{a,b,c}</math> is incident to <math>2^a+2^b+2^c-3</math> [[line\|combinatorial lines]].

	The [[equal-slices measure]] gives each slice a total measure of 1.		The [[equal-slices measure]] on <math>[3]^n</math> gives each slice a total measure of <math>1</math>.

	The k=2 analogue of a slice is sometimes ~~known as~~ a ''layer'', ~~and~~ plays a role in the proof of [[Sperner's theorem]].		The <math>k=2</math> analogue of a slice, sometimes called a ''layer'', plays a role in the proof of [[Sperner's theorem]].

	A [[combinatorial line]] must touch three slices <math>\Gamma_{a+r,b,c}, \Gamma_{a,b+r,c}, \Gamma_{a,b,c+r}</math> in an equilateral triangle. Conversely, one might hope that any sufficiently "rich" subsets of three slices <math>\Gamma_{a+r,b,c}, \Gamma_{a,b+r,c}, \Gamma_{a,b,c+r}</math> in an equilateral triangle will have many combinatorial lines between them. However there appear to be quite a lot of obstructions to this hope, consider e.g. the subset of <math>\Gamma_{a+r,b,c}</math> of strings whose first digit is 1, and the subsets of <math>\Gamma_{a,b+r,c}, \Gamma_{a,b,c+r}</math> of strings whose first digit is 2.		A [[combinatorial line]] must touch three slices <math>\Gamma_{a+r,b,c}, \Gamma_{a,b+r,c}, \Gamma_{a,b,c+r}</math> corresponding to an equilateral triangle in <math>\Delta_n</math>. Conversely, one might hope that any sufficiently "rich" subsets of three slices <math>\Gamma_{a+r,b,c}, \Gamma_{a,b+r,c}, \Gamma_{a,b,c+r}</math> in an equilateral triangle will have many combinatorial lines between them. However there appear to be quite a lot of obstructions to this hope, consider e.g. the subset of <math>\Gamma_{a+r,b,c}</math> of strings whose first digit is 1, and the subsets of <math>\Gamma_{a,b+r,c}, \Gamma_{a,b,c+r}</math> of strings whose first digit is 2.

Gowers: added internal link

2009-02-18T09:48:47Z

added internal link

← Older revision		Revision as of 02:48, 18 February 2009
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	The cube <math>[3]^n</math> has <math>\frac{(n+1)(n+2)}{2}</math> slices, and each slice <math>\Gamma_{a,b,c}</math> has cardinality <math>\frac{n!}{a!b!c!}</math>.		The cube <math>[3]^n</math> has <math>\frac{(n+1)(n+2)}{2}</math> slices, and each slice <math>\Gamma_{a,b,c}</math> has cardinality <math>\frac{n!}{a!b!c!}</math>.

	Each slice <math>\Gamma_{a,b,c}</math> is incident to <math>2^a+2^b+2^c-3</math> combinatorial lines.		Each slice <math>\Gamma_{a,b,c}</math> is incident to <math>2^a+2^b+2^c-3</math> [[line\|combinatorial lines]].

	The [[equal-slices measure]] gives each slice a total measure of 1.		The [[equal-slices measure]] gives each slice a total measure of 1.

Teorth at 22:51, 15 February 2009

2009-02-15T22:51:12Z

← Older revision		Revision as of 15:51, 15 February 2009
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	The k=2 analogue of a slice is sometimes known as a ''layer'', and plays a role in the proof of [[Sperner's theorem]].		The k=2 analogue of a slice is sometimes known as a ''layer'', and plays a role in the proof of [[Sperner's theorem]].

			A [[combinatorial line]] must touch three slices <math>\Gamma_{a+r,b,c}, \Gamma_{a,b+r,c}, \Gamma_{a,b,c+r}</math> in an equilateral triangle. Conversely, one might hope that any sufficiently "rich" subsets of three slices <math>\Gamma_{a+r,b,c}, \Gamma_{a,b+r,c}, \Gamma_{a,b,c+r}</math> in an equilateral triangle will have many combinatorial lines between them. However there appear to be quite a lot of obstructions to this hope, consider e.g. the subset of <math>\Gamma_{a+r,b,c}</math> of strings whose first digit is 1, and the subsets of <math>\Gamma_{a,b+r,c}, \Gamma_{a,b,c+r}</math> of strings whose first digit is 2.

Teorth at 22:08, 15 February 2009

2009-02-15T22:08:17Z

← Older revision		Revision as of 15:08, 15 February 2009
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	The [[equal-slices measure]] gives each slice a total measure of 1.		The [[equal-slices measure]] gives each slice a total measure of 1.

	The k=2 analogue of a slice is sometimes known as a ''layer''.		The k=2 analogue of a slice is sometimes known as a ''layer'', and plays a role in the proof of [[Sperner's theorem]].

Teorth: New page: A '''slice''' of the cube $[3]^n$ is a set of the form $\Gamma_{a,b,c}$ for some (a,b,c) in the triangular grid : $\Delta_n := \{ (a,b,c) \in {\Bbb Z}_+^3: a+b+c...$

2009-02-15T22:07:57Z

New page: A '''slice''' of the cube <math>[3]^n</math> is a set of the form <math>\Gamma_{a,b,c}</math> for some (a,b,c) in the triangular grid :<math>\Delta_n := \{ (a,b,c) \in {\Bbb Z}_+^3: a+b+c...

New page

A '''slice''' of the cube <math>[3]^n</math> is a set of the form <math>\Gamma_{a,b,c}</math> for some (a,b,c) in the triangular grid

:<math>\Delta_n := \{ (a,b,c) \in {\Bbb Z}_+^3: a+b+c = n \},</math>

where <math>\Gamma_{a,b,c}</math> is the set of all strings with a 1s, b 2s, and c 3s. Thus for instance

:<math>\Gamma_{1,2,0} = \{ 011, 101, 101 \}.</math>

The cube <math>[3]^n</math> has <math>\frac{(n+1)(n+2)}{2}</math> slices, and each slice <math>\Gamma_{a,b,c}</math> has cardinality <math>\frac{n!}{a!b!c!}</math>.

Each slice <math>\Gamma_{a,b,c}</math> is incident to <math>2^a+2^b+2^c-3</math> combinatorial lines.

The [[equal-slices measure]] gives each slice a total measure of 1.

The k=2 analogue of a slice is sometimes known as a ''layer''.