Ramsey-Collatz Correlation and Some Extremal Combinatorics, along with Ramsey-Mahler Rare Concurrencies

For a ternary tree T of depth n with 0-1 labeled edges, its weight f (T ) is the least number of path labels among binary subtrees. The maximum f (n), over all labelings, of these weights starts with 1,2,3,4,8. We show f (6) ≥ 12, but focus on depth 5 (with 2³⁶³ trees). We approximate the percentages for weights 1–8: 0, 1.04, 23.6, 55.0, 18.8, 1.54, 0, 0; our linked supplements include thousands of mined trees of rare weights 7-8. Our next products additionally relate to Mahler’s 32 {3 \over 2} -problem and large stopping times showing a real number is not a Z-number. Our version is iterated multiplication of integers by 23 {2 \over 3} . For a certain sequence of integer intervals, we present a choice function. Our interval I_g has left endpoint a(g)=min{k|{g⋅(23)k}≥12∨k=⌈log32(g)⌉} a(g) = \min \left\{ k \mid \left\{ g \cdot \left(\frac{2}{3}\right)^k \right\} \ge \frac{1}{2} \;\vee\; k = \left\lceil \log_{\frac{3}{2}}(g) \right\rceil \right\} , and right endpoint at the stopping time for “the 1st intermediate rounding error”, but the interpolation is a “no sudden death” function. We present simultaneous peculiarity in base 2 and base 3: numbers g with large values (for the size of g)of a(g) and length of I_g, which also have a less common Ramsey weight (when written in base 2 and used to edge-label a tree), or are prime. Then we cross the weight notion with the Collatz scaled total stopping time γ_∞(n). We construct sizable low-high sequences of 8-tuples of same-weight pairs of numbers below 2³⁶³ with certain monotonicity in values of γ_∞, and in six apartness levels > 10⁻ⁱ for i =1,..., 6. Apartness of γ_∞-values would be in the ‘low’ and ‘high’ halves as well as between the corresponding components of terms of the sequence. We get lower bounds for their lengths, and for Collatz-landing-apart just for weight 8. We statistically establish an unexpected correlation between Collatz and Ramsey.