Distinguishing Number of Fano Plane?

Question

I'm trying to find an exact distinguishing number for the fano plane. Through trial I've got it to D(Fano) $\leq 4$.

Any ideas?

Answer 1

The Fano plane can be labeled with triples $(a,b,c)$ of three coordinates in $\mathbb{Z_2}=\{0,1\}$, where the labellings on the seven points are all choices other than $(0,0,0).$ Then three points are collinear iff their sum is $(0,0,0).$ In this way the Fano plane is the set of nonzero elements of the additive group $\mathbb{Z}_2^3$. One usual labeling puts the generators $(1,0,0),(0,1,0),(0,0,1)$ on the vertices of the "big triangle", thereafter adding endpoints to get the third point on each line. Then the point in the center of the triangle winds up as $(1,1,1).$

The automorphism group of the Fano plane then consists of all $3 \times 3$ matrices over $\mathbb{Z}_2$ whose determinant is nonzero. So if you color the generators $(1,0,0),(0,1,0),(0,0,1)$ as say red, yellow blue, and color all other points black, then any automorphism fixing the colors must be the identity automorphism, since it fixes the generators. [For example since $(1,1,0)=(1,0,0)+(0,1,0)$, and the latter two are fixed by an automorphism $f$, the point $(1,1,0)$ must also be fixed by $f$.

So this shows that you can distinguish using four colors. Thus one may conclude that $D \le 4$ as you say. To show it is not less than 4, one thing I noticed is that with 3 colors there would be 7 things placed in 3 bins, so that at least three have the same color. However the structure of the automorphism group is complicated, and probably some use of that structure would be necessary to rule out a distinguishing coloring using only 3 (or fewer) colors.