Pinning–the very start, cnt 2 (Certain 2-colorings)

It’s Sunday, h21:19. Starbucks. I’ll have to leave in less than 40m. The time is passing by awfully fast.

I am looking for the 2-colorings of triangles, which avoid unicolor tetrahedra. I’ll try the colorings $\gamma := F \circ \sigma$ which are split into a composition of

$\sigma : \binom X3 \rightarrow \{3\ \ldots\ 3\cdot(n-1)\}$

and

$F : \{3\ \ldots\ 3\cdot(n-1)\} \rightarrow \{0\ 1\}$

where

$X := \{0\ \ldots\ n\!-\!1\}$

and

$\sigma(a\ b\ c) := a+b+c$

Then for every tetrahedron $T := \{a\ b\ c\ d\}$ , and $\tau := a+b+c+d$ , we obtain the following $\sigma$ -values of its walls

$T_x := T\setminus\{x\}$

namely:

$\sigma(T_x) = \tau - x$

Assume $a < b < c < d$ . Then

$\sigma(T_a) + \sigma(T_b) + \sigma(T_x) + \sigma(T_x) = 3\cdot \tau$ (is divisible by 3)

$3 \le \sigma(T_d) < \sigma(T_c) < \sigma(T_b) < \sigma(T_a) \le 3\cdot(n-2)$

$n > \sigma(T_a) - \sigma(T_d) \ge 3$

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Pinning–the very start, cnt 2 (Certain 2-colorings)

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