general.hpp File Reference

Plans regarding van der Waerden problems. More...

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Detailed Description

Plans regarding van der Waerden problems.

Relations to other modules
Handling of duplicate entries for vdW-numbers
  • 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.
Lower bounds
Considering finite commutative groups G
  • For a natural number k >= 1 assume that every non-null element of G has order at least k. Then one can define (non-degenerated) 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 non-null 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 m-labelling there exists i such that colour-class i is independent w.r.t. k_i-progressions.
  • 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}.
Automorphisms of van-der-Waerden hypergraphs
  • We need to investigate experimentally the symmetries of these hypergraphs.
Automorphisms of van-der-Waerden clause-sets
  • The obvious automorphisms of diagonal van-der-Waerden clause-sets are given by the inner product of automorphisms for the underlying van-der-Waerden hypergraph and the S_m when using m parts.
  • Are there more?

Definition in file general.hpp.