Quelltext der Seite Benutzer:Yuriklim/Satz von Hales-Jewett

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In [[mathematics]], the '''Hales–Jewett theorem''' is a fundamental [[Combinatorics|combinatorial]] result of [[Ramsey theory]] named after [[Alfred W. Hales]] and Robert I. Jewett, concerning the degree to which high-dimensional objects must necessarily exhibit some combinatorial structure; it is impossible for such objects to be "completely random".<ref>{{cite journal|first1=Alfred W.|last1=Hales|first2=Robert I.|last2=Jewett|title=Regularity and positional games|journal=[[Trans. Amer. Math. Soc.]]|volume=106|issue=2|year=1963|pages=222–229|mr=0143712|doi=10.1090/S0002-9947-1963-0143712-1|doi-access=free}}</ref>

An informal geometric statement of the theorem is that for any positive integers ''n'' and ''c'' there is a number ''H'' such that if the cells of a ''H''-dimensional ''n''×''n''×''n''×...×''n'' cube are colored with ''c'' colors, there must be one row, column, or certain diagonal (more details below) of length ''n'' all of whose cells are the same color. In other words, the higher-dimensional, multi-player, ''n''-in-a-row generalization of a game of [[tic-tac-toe]] cannot end in a draw, no matter how large ''n'' is, no matter how many people ''c'' are playing, and no matter which player plays each turn, provided only that it is played on a board of sufficiently high dimension ''H''.  By a standard [[Strategy stealing|strategy stealing argument]], one can thus conclude that if two players alternate, then the first player has a winning strategy when ''H'' is sufficiently large, though no practical algorithm for obtaining this strategy is known.

More formally, let ''W''<sub>''n''</sub><sup>''H''</sup> be the set of words of length ''H'' over an alphabet with ''n'' letters; that is, the set of sequences of  {1, 2, ..., ''n''} of length ''H''.  This set forms the hypercube that is the subject of the theorem.   
A ''variable word'' ''w''(''x'') over ''W''<sub>''n''</sub><sup>''H''</sup> still has length ''H'' but includes the special element ''x'' in place of at least one of the letters.  The words ''w''(1), ''w''(2), ..., ''w''(''n'') obtained by replacing all instances of the special element ''x'' with 1, 2, ..., ''n'', form a [[Combinatorial cube|combinatorial line]] in the space ''W''<sub>''n''</sub><sup>''H''</sup>; combinatorial lines correspond to rows, columns, and (some of the) diagonals of the [[hypercube]].  The Hales–Jewett theorem then states that for given positive integers ''n'' and ''c'', there exists a positive integer ''H'', depending on ''n'' and ''c'', such that for any partition of ''W''<sub>''n''</sub><sup>''H''</sup> into ''c'' parts, there is at least one part that contains an entire combinatorial line.

For example, take ''n'' = 3, ''H'' = 2, and ''c'' = 2.  The hypercube ''W''<sub>''n''</sub><sup>''H''</sup> in this case
is just the standard [[tic-tac-toe]] board, with nine positions:

{| 
|11
|12
|13
|- 
|21
|22
|23
|-
|31
|32
|33
|}

A typical combinatorial
line would be the word 2x, which corresponds to the line 21, 22, 23; another combinatorial line is xx, which is the line
11, 22, 33.  (Note that the line 13, 22, 31, while a valid line for the game [[tic-tac-toe]], is not considered a combinatorial line.)  In this particular case, the Hales–Jewett theorem does not apply; it is possible to divide
the [[tic-tac-toe]] board into two sets, e.g. {11, 22, 23, 31} and {12, 13, 21, 32, 33}, neither of which contain
a combinatorial line (and would correspond to a draw in the game of [[tic-tac-toe]]).  On the other hand, if we increase
''H'' to, say, 8 (so that the board is now eight-dimensional, with 3<sup>8</sup> = 6561 positions), and partition this board
into two sets (the "noughts" and "crosses"), then one of the two sets must contain a combinatorial line (i.e. no draw is possible in this variant of [[tic-tac-toe]]).  For a proof, see below.

== Proof of Hales–Jewett theorem (in a special case) ==

We now prove the Hales–Jewett theorem in the special case ''n''&nbsp;=&nbsp;3, ''c''&nbsp;=&nbsp;2, ''H''&nbsp;=&nbsp;8 discussed above.  The idea is to
reduce this task to that of proving simpler versions of the Hales–Jewett theorem (in this particular case, to the cases ''n''&nbsp;=&nbsp;2, ''c''&nbsp;=&nbsp;2, ''H''&nbsp;=&nbsp;2 and ''n''&nbsp;=&nbsp;2, ''c''&nbsp;=&nbsp;6, ''H''&nbsp;=&nbsp;6).  One can prove the general case of the Hales–Jewett theorem by similar methods, using [[mathematical induction]].

Each element of the [[hypercube]] ''W''<sub>3</sub><sup>8</sup> is a string of eight numbers from 1 to 3, e.g. 13211321 is an element of the [[hypercube]].  We are assuming that this [[hypercube]] is completely filled with "noughts" and "crosses".  We shall use a [[proof by contradiction]] and assume that neither the set of noughts nor the set of crosses contains a combinatorial line.  If we fix the first six elements of such a string and let the last two vary, we obtain an ordinary [[tic-tac-toe]] board, for instance 132113?? gives such a board.  For each such board abcdef??, we consider the positions
abcdef11, abcdef12, abcdef22.  Each of these must be filled with either a nought or a cross, so by the [[pigeonhole principle]] two of them must be filled with the same symbol.  Since any two of these positions are part of
a combinatorial line, the third element of that line must be occupied by the opposite symbol (since we are assuming that no combinatorial line has all three elements filled with the same symbol).  In other words, for each choice of abcdef
(which can be thought of as an element of the six-dimensional hypercube ''W''<sub>3</sub><sup>6</sup>), there are six (overlapping) possibilities:

# abcdef11 and abcdef12 are noughts; abcdef13 is a cross.
# abcdef11 and abcdef22 are noughts; abcdef33 is a cross.
# abcdef12 and abcdef22 are noughts; abcdef32 is a cross.
# abcdef11 and abcdef12 are crosses; abcdef13 is a nought.
# abcdef11 and abcdef22 are crosses; abcdef33 is a nought.
# abcdef12 and abcdef22 are crosses; abcdef32 is a nought.

Thus we can partition the six-dimensional [[hypercube]] ''W''<sub>3</sub><sup>6</sup> into six classes, corresponding to each of the above six possibilities.  (If an element abcdef obeys multiple possibilities, we can choose one arbitrarily, e.g. by choosing the highest one on the above list).

Now consider the seven elements 111111, 111112, 111122, 111222, 112222, 122222, 222222 in ''W''<sub>3</sub><sup>6</sup>.  By the [[pigeonhole principle]], two of these elements must fall into the same class.  Suppose for instance
111112 and 112222 fall into class (5), thus 11111211, 11111222, 11222211, 11222222 are crosses and 11111233, 11222233 are noughts.  But now consider the position 11333233, which must be filled with either a cross or a nought.  If it is filled with a cross, then the combinatorial line 11xxx2xx is filled entirely with crosses, contradicting our hypothesis.  If instead it is filled with a nought, then the combinatorial line 11xxx233 is filled entirely with noughts, again contradicting our hypothesis.  Similarly if any other two of the above seven elements of ''W''<sub>3</sub><sup>6</sup> fall into the same class.  Since we have a contradiction in all cases, the original hypothesis must be false; thus there must exist at least one combinatorial line consisting entirely of noughts or entirely of crosses.

The above [[argument]] was somewhat wasteful; in fact the same theorem holds for ''H''&nbsp;=&nbsp;4.<ref>{{cite journal|first1=Neil|last1=Hindman|first2=Eric|last2=Tressler|url=http://nhindman.us/research/hjnumb.pdf|title=The first nontrivial Hales-Jewett number is four|journal=[[Ars Combinatoria (journal)|Ars Combinatoria]]|year=2014|volume=113|pages=385–390|mr=3186481}}</ref>
If one extends the above argument to general values of ''n'' and ''c'', then ''H'' will grow very fast; even when ''c''&nbsp;=&nbsp;2 (which corresponds to two-player [[tic-tac-toe]]) the ''H'' given by the above argument grows as fast as the [[Ackermann function]].  The first [[primitive recursive]] bound is due to [[Saharon Shelah]],<ref>{{cite journal|first=Saharon|last=Shelah|title=Primitive recursive bounds for van der Waerden numbers|journal=[[J. Amer. Math. Soc.]]|volume=1|year=1988|pages=683–697|issue=3|doi=10.2307/1990952|mr=929498|jstor=1990952|doi-access=free}}</ref> and is still the best known bound in general for the ''Hales–Jewett number'' ''H''&nbsp;=&nbsp;''H''(''n'',&nbsp;''c'').

== Connections with other theorems ==
Observe that the above argument also gives the following corollary: if we let ''A'' be the set of all
eight-digit numbers whose digits are all either 1, 2, 3 (thus ''A'' contains numbers such as 11333233),
and we color ''A'' with two colors, then ''A'' contains at least one [[arithmetic progression]] of length three, all of whose elements are the same color.  This is simply because all of the combinatorial lines appearing in the above proof of the Hales–Jewett theorem, also form arithmetic progressions in [[Decimal|decimal notation]]. A more general formulation of this argument can be used to show that the Hales–Jewett theorem generalizes [[van der Waerden's theorem]]. Indeed the Hales–Jewett theorem is substantially a stronger theorem.

Just as [[van der Waerden's theorem]] has a stronger ''density version'' in [[Szemerédi's theorem]], the Hales–Jewett theorem also has a density version. In this strengthened version of the Hales–Jewett theorem, instead of coloring the entire [[hypercube]] ''W''<sub>''n''</sub><sup>''H''</sup> into ''c'' colors, one is given an arbitrary subset ''A'' of the hypercube ''W''<sub>''n''</sub><sup>''H''</sup> with some given density 0 < δ < 1.  The theorem states that if ''H'' is  sufficiently large depending on ''n'' and δ, then the set ''A'' must necessarily contain an entire combinatorial line.

The density Hales–Jewett theorem was originally proved by Furstenberg and Katznelson using [[ergodic theory]].<ref>{{cite journal|last1=Furstenberg|first1=Hillel|last2=Katznelson|author1link=Hillel Furstenberg|first2=Yitzhak|author2link=Yitzhak Katznelson|title=A density version of the Hales–Jewett theorem|journal=Journal d'Analyse Mathématique|volume=57|year=1991|pages=64–119|mr=1191743|issue=1|doi=10.1007/BF03041066}}</ref>  In 2009, the [[Polymath Project]] developed a new proof<ref>{{cite journal|title=A new proof of the density Hales–Jewett theorem|author=D. H. J. Polymath|journal=[[Annals of Mathematics]]|year=2012|pages=1283–1327|volume=175|issue=3|mr=2912706|doi=10.4007/annals.2012.175.3.6|doi-access=free}}</ref><ref>{{cite encyclopedia|chapter=Polymath and the density Hales-Jewett theorem|publisher=János Bolyai Mathematical Society|editor2-last=Solymosi|editor2-first=József|editor1-last=Bárány|editor1-first=Imre|isbn=978-963-9453-14-2|location=Budapest|doi=10.1007/978-3-642-14444-8_21|last=Gowers|year=2010|title=An irregular mind|pages=659–687|volume=21|series=Bolyai Society Mathematical Studies|mr=2815619|first=William Timothy|editor2-link=József Solymosi}}</ref> of the density Hales–Jewett theorem based ideas on from the proof of the [[corners theorem]].<ref>{{cite journal|last1=Ajtai|first1=Miklós|last2=Szemerédi|first2=Endre|title=Sets of lattice points that form no squares|journal=Stud. Sci. Math. Hungar.|volume=9|pages=9–11|year=1974|mr=0369299}}</ref> Dodos, Kanellopoulos, and Tyros gave a simplified version of the Polymath proof.<ref>{{cite journal|title=A simple proof of the density Hales–Jewett theorem|last1=Dodos|first1=Pandelis|last2=Kanellopoulos|first2=Vassilis|last3=Tyros|first3=Konstantinos|journal=[[Int. Math. Res. Not. IMRN]]|year=2014|volume=2014|issue=12|pages=3340–3352|mr=3217664|doi=10.1093/imrn/rnt041|arxiv=1209.4986}}</ref>

The Hales–Jewett is generalized by the [[Graham–Rothschild theorem]], on higher-dimensional [[Combinatorial cube|combinatorial cubes]].

== See also ==

* [[Bartel Leendert van der Waerden]]

== References ==
<references />
{{Tic-Tac-Toe}}

== External links ==

* [https://www.ti.inf.ethz.ch/ew/courses/extremal04/razen.pdf Full proof of HJT] - begins on slide 57
* [http://www.sciencenews.org/view/generic/id/50532/title/Mathematics_by_collaboration Science News article on the collaborative proof of the density Hales-Jewett theorem]
* [http://www.thebigquestions.com/2010/04/08/blogging-tic-tac-toe-and-the-future-of-math/ A blog post by Steven Landsburg discussing how the proof of this theorem was improved collaboratively on a blog]