Plans regarding van der Waerden problems.
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Detailed Description
Plans regarding van der Waerden problems.
 Todo:
 Relations to other modules
 Todo:
 Handling of duplicate entries for vdWnumbers

The functions (pd)vanderwaerdenxk, (pd)vanderwaerdend2 etc. overlap in their stored values.

Perhaps it is best to remove these duplications.

On the other hand, it might be a burden to plan for all cases, and one might just write a test which checks for consistency.
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 Lower bounds
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 Considering finite commutative groups G

For a natural number k >= 1 assume that every nonnull element of G has order at least k. Then one can define (nondegenerated) arithmetic progressions of length k as sets {a + i*d : 0 <= i < k} for a, d in G, d <> 0.


Let the hypergraph arithprog_hg_cgrp(k,G) have vertices the elements of G, while the hyperedges are the arithmetic progressions of length k.

So for a fixed G one can consider the transversal and independence number of arithprog_hg_cgrp(k,G) as well as the chromatic number.

In order to consider sequences, one possibility is to consider powers G^n for some fixed G.

It seems that in this context only the transversal and independence numbers are considered; however we should also consider the chromatic number. For a given minimal order mo(G) of the nonnull elements we can consider all 1 <= k <= mo(G).

For G = ZZ_n we can also define vanderwaerden_m^mod(k_1, ..., k_m) as the smallest n >= max(k_1,..,k_m) such that for every mlabelling there exists i such that colourclass i is independent w.r.t. k_iprogressions.

If G is infinite, then we can consider an enumeration of the elements, obtaining a generalisation of vanderwaerden_m(k_1,...,k_m) in this way (allowing then for given n only arithmetic progressions which lie inside the first n elements).

Apparently alpha_arithprog_hg_cgrp(k,G) and especially n > alpha_arithprog_hg_cgrp(k,G^n) are only considered in the literature.

And this especially for G in {ZZ_3, ZZ_5}.
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 Automorphisms of vanderWaerden hypergraphs

We need to investigate experimentally the symmetries of these hypergraphs.
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 Automorphisms of vanderWaerden clausesets

The obvious automorphisms of diagonal vanderWaerden clausesets are given by the inner product of automorphisms for the underlying vanderWaerden hypergraph and the S_m when using m parts.

Are there more?
Definition in file general.hpp.