PHP and Ramsey Theory

1) Pigeonhole Principle (PHP): There will be at least one repeat of a dozen in 4 spins (excluding zero(es)), i.e. one pigeonhole will contain two pigeons: 1323 or 121. Each pigeon, in this basic dozens example, is equally-likely yet still possessing some default characteristics based on their relationship to other pigeons and underlying event structures, but advanced PHP can render pigeons NOT equally-likely. All subsequent theorems of the Ramsey flavour are entirely built on PHP in terms of multiple repeats over slightly larger limits (except for Erdos; describes structures relating to uniques instead of repeats):

2) Van der Waerden's Theorem (VdW): In 9 trials in Roulette yielding 2 colours, parities or partitions, there will be one Arithmetic Progression (AP) of 3 integers holding the same value with equal distance, i.e. 1,2,3 (consecutive) or 1,5,9 (non-consecutive). There are 16 possible APs in total. Within 9 spins there could be a single AP - or there could be many - but there has to be at least one as a mathematical certainty.

3) Theorem on Friends and Strangers (Friends'): In a group of 6 lines (acting as people) there will be at least one trio the same (friends) or at least one trio different (strangers). Examples: 211134 = 1 trio same + 1 trio different; 111134 = at least 1 trio same; 113246 = at least 1 trio different.

4) Erdős–Szekeres theorem (Erdos): In 3 unique dozens there will be an increasing sub-sequence or decreasing sub-sequence, i.e. 1,2,3 has an increasing sub-sequence consisting of all three numbers; 1,3,2 has a decreasing sub-sequence of 3,2; 3,1,2 has two decreasing sub-sequences, 3,1 and 3,2; 3,2,1 has three decreasing length-2 sub-sequences, 3,2, 3,1, and 2,1. This shows that some permutations involving unique numbers are rarer than others by virtue of this mathematical law; PHP and Ramsey theorems, in turn, are built upon this as the most fundamental law.

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