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In the first post: Solvability of a system related to the subsets of {1,2,3}, we have shown (here) that an allowed labeling $f$ of $B_3$ can have a negative Euler totient $\varphi(f)$ [equals for example to $-1/4 + 3/100$ for $f$ below].

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We have also proved that for any allowed labeling $f$ of $B_3$ then $\varphi(f) > -1$.

So the natural question is now the following:
Question: What's the infimum of $\{ \varphi(f) \mid f$ an allowed labeling $f$ of $B_3 \}$? Is it $-1/4$? What's the proof?

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According to Mathematica, this infimum is exactly $-1/4$:

In[1]:= FindMinimum[{x1 - x2 - x3 - x4 + x5 + x6 + x7 - 1, x1 > x2 && x1 > x3 && x1 > x4 && x2 > x5 && x2 > x6 && x3 > x5 && x3 > x7 && x4 > x6 && x4 > x7 && x5 > 1 && x6 > 1 && x7 > 1 && x2 x3 <= x1 x5 && x2 x4 <= x1 x6 && x3 x4 <= x1 x7 && x5 x6 <= x2 && x5 x7 <= x3 && x6 x7 <= x4}, {x1, x2, x3, x4, x5, x6, x7}]

Out[1]= {-0.25, {x1 -> 2.25, x2 -> 1.5, x3 -> 1.5, x4 -> 1.5, x5 -> 1., x6 -> 1., x7 -> 1.}}

This infimum is not realized but the answer of the previous post
https://math.stackexchange.com/a/1950465/84284
gives a sequence $f_n$ such that $\varphi(f_n)$ converges to the infimum: take $x=9/4$ and $\epsilon_n = 1/n$.

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