Moser's cube problem

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Let [math]c'_n[/math] denote the largest subset of [math][3]^n[/math] which does not contain any geometric line (which is the same as a combinatorial line, but has a second wildcard y which goes from 3 to 1 whilst x goes from 1 to 3, e.g. xx2yy gives the geometric line 11233, 22222, 33211). The Moser cube problem is to understand the behaviour of [math]c'_n[/math]. The first few values are (see OEIS A003142):

[math]c'_0 = 1; c'_1 = 2; c'_2 = 6; c'_3 = 16; c'_4 = 43.[/math]

The best known lower bound for [math]c'_n[/math] is

[math]c'_n \gg 3^n/\sqrt{n}[/math],

formed by fixing the number of 2s to a single value near n/3. Is it possible to do any better? Note that we have a [upper and lower bounds|significantly better bound] for [math]c_n[/math]:

[math]c'_n \geq 3^{n-O(\sqrt{\log n})}[/math].