Plans regarding palindromic van der Waerden numbers.
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Detailed Description
Plans regarding palindromic van der Waerden numbers.
 Todo:
 Palindromic numbers

As discussed in "Palindromic versions" in Experimentation/Investigations/RamseyTheory/VanderWaerdenProblems/plans/general.hpp we should provide the smallest n_1 such that for n' >= n_1 no palindromic solution exists, and the largest n_0, such that for all n' <= n_0 palindromic solutions do exist.

So the "palindromic number" is actually a pair of numbers (namely (n_0,n_1)).

Naming:

pdvanderwaerden(t)

How to access the parts (sat resp. unsatrelated)? At Maximalevel we could simply use pdvanderwaerden(t)[1 / 2]. It works, however it is not very expressive.

DONE (palindromic versions are something special, a kind of "universal variation", and so they deserve their own treatmeant) Or perhaps we should provide g_vanderwaerden(t,p), where p is some parameter, perhaps a string, which selects which form of a generalised vdWnumber is considered.

DONE (not chosen) Or perhaps vanderwaerden_g(t,p), in this case vanderwaerden_g(t,"pd"). This seems best.

How to name this quantity in mathematical formulas?

DONE (not chosen) "pdvdw_m(t)"

"vdw_m^{pd}(t)": this seems better, since it is a variation on vdWnumbers; and in general variations of Ramseytype numbers are indicated by an upper index.

Note that this "number" is actually a pair.

How to access the two components of the pairs? Lower and upper index are already used. One could write "vdw_m^{pd}(t)_{1 / 2}" (index 1 for the satpart, index 2 for the unsatpart) ? Correct, but not very expressive.

In a context where "w(m; t_1,..., t_m)" is used for "vdw_m(t)", one can use "w^{pd}(m; t_1,...,t_m)" for vdw_m^pd(t), and "w^{pd}_{1 / 2}(m; t_1,...,t_m)" for the sat resp. unsatcomponent.

Again, indices "1", "2" are not much telling. "0", "1" for "unsat", "sat" would be better, however then we have a clash with the ordering of the components (and their real indices).

From a solution for n a solution for n2 can be derived by removing vertices n and 1, and then subtracting 1 from the remaining vertices.

Thus the sequence of bits for n=1,... (always for a given tuple t) is first all 1's, then 0 and 1 strictly alternating, and then all 0,s (where the lenght of the alternating sequence could be zero).

DONE (integrated) Dual versions (regarding satisfiability):

So we also need to consider the largest n such that all n' <= n yield satisfiable problems.

At the Maximaplevel we could name that vanderwaerden_g(t,"pdsat").

In mathematical formulas we could use "svdw_m^{pd}(t)".

The two numbers svdw_m^{pd}(t), vdw_m^{pd}(t) completely determine satisfiability for all n (for this t).
Definition in file PdNumbers.hpp.