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General plans regarding investigations on Ramsey theory (Ramsey problems, van der Waerden problems, etc.) More...

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General plans regarding investigations on Ramsey theory (Ramsey problems, van der Waerden problems, etc.)

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Create milestones.
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Connections
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Hales-Jewett problems
• The length k of arithmetic progressions is now called t.
• And N now stands for the "dimension" of the set of vertices, while its size is n = t^N.
• And the vertices are not arbitrary elements but the elements of V_t^N := {1,...,t}^N.
• Instead of an arithmetic progression now "lines" are considered, which are t-tuples of elements of V such that for each coordinate we have a possibly degenerated (ascending) arithmetic progression ("degenerated" allows slope 0), where for at least one coordinate the arithmetic progression (which must be just (1,...,t)) is non-degenerated.
• So the hypergraphs are (V_t^N, E_t^N), where the hyperedges are the t-subsets of V such that an ordering exists making this subset to a "line" (such an ordering is then unique).
• We have |E_t^N| = sum_{i=0}^{t-1} binomial(t,i) * N^i, where i stands for the number of degeneration-coordinates.
• The Hales-Jewett theorem now asserts the existence of halesjewett_r(t) = N, so that N' >= N is equivalent to the hypergraph (V_t^N, E_t^N) not being r-colourable.
• We have vanderwaerden_r(t) <= t^halesjewett_r(t), since using the bijection from {1,...,n} to V_t^N given by interpreting the elements of V_t^n as base-t-representation of natural numbers, but where we have to subtract 1 from each such digit, lines yield special arithmetic progressions.
• It seems not possible to create natural "mixed forms", since for different line-lengths t we have to use different vertices (namely tuples over {1,..,t}).
• On the other hand, using the notion of arithmetic progression as we did it, one could for example consider arithmetic progressions in a base set {1,...,T} with T = max {t_1,...,t_r} of slope 0 or 1 (i.e., in each coordinate we must have such an arithmetic progression, and where at least for one coordinate the slope is 1).
• There is a generalisation halesjewett_r^d(t), where d=1 for the above form, and where instead of lines d-dimensional "subspaces" are considered.
• It is known that halesjewett_r(2) = r.
• Likely we should create a new module ComputerAlgebra/RamseyTheory/Lisp/HalesJewett.
• There is a project about Hales-Jewett numbers: http://michaelnielsen.org/polymath1/index.php
1. So there is actually considerable interest in computing Hales-Jewett numbers!
2. We have http://www.math.ucsd.edu/~etressle/hj32.pdf, where halesjewett_2(3) = 4 is shown, directly with a proof by case distinctions, and mentioning also an algorithm.
3. At http://michaelnielsen.org/polymath1/index.php?title=Hales-Jewett_theorem we find likely the most up-to-date bounds.
4. halesjewett_r(3):
1. r=3: > 13
2. r=4: > 37
3. r=5: > 84
4. r=6: > 103
5. These case use vanderwaerden_r(3).
5. halesjewett_r(4):
1. r=2: > 11
2. r=3: > 97
3. r=4: > 349
4. r=5: > 751
5. r=6: > 3259
6. These case use vanderwaerden_r(4).
6. halesjewett_r(5):
1. r=2: > 59
2. r=3: > 302
3. r=4: > 2609
4. r=5: > 6011
5. r=6: > 14173
6. These case use vanderwaerden_r(5).
7. halesjewett_r(6):
1. r=2: > 226
2. r=3: > 1777
3. r=4: > 18061
4. r=5: > 49391
5. r=6: > 120097
6. These case use vanderwaerden_r(6).
8. halesjewett_r(7):
1. r=2: > 617
2. r=3: > 7309
3. r=4: > 64661
4. These case use vanderwaerden_r(7).
9. halesjewett_r(8):
1. r=2: > 1069
2. r=3: > 34057
3. These case use vanderwaerden_r(8).
10. halesjewett_r(9):
1. r=2: > 3389
2. These case use vanderwaerden_r(9).
11. Also the density considerations are of interest, since the hypergraphs sequences have the density property, i.e., the quotients alpha_halesjewett_hg(k,N)) / k^N -> 0 for fixed k.
12. For k=3 these independence numbers are studied at http://michaelnielsen.org/polymath1/index.php?title=Upper_and_lower_bounds where c_n = alpha_halesjewett_hg(3,n)) is used.
13. Precise values are known for 0 <= n <= 6: 1,2,6,18,52,150,450.
14. A simple greedy algorithm is given there, but his works only up to n <= 3.
15. Apparently their main algorithm is a genetic one: http://michaelnielsen.org/polymath1/index.php?title=Genetic_algorithm
16. We must really work on hypergraph transversal algorithms!
17. The general case is considered at http://michaelnielsen.org/polymath1/index.php?title=Higher-dimensional_DHJ_numbers where the notation c_{n,k} = alpha_halesjewett_hg(k,n)) is used.
18. A simple upper bound is c_{n,k} <= (k-1)*k^{n-1}; parameter pairs where this upper bound is attained are called "saturated".

Definition in file general.hpp.