# On 'Very Movable' Geometric Configurations (Configurations with a large degree of freedom)

Let $C$ be an $(n_r, b_k)$ combinatorial configuration that admits a geometric realization in the plane. I'm interested in the maximum number of points/lines $M$ of $C$ we can place in general position on the plane, before the remainder of the configuration is completely determined. Alternatively, given a geometric embedding, how many degrees of freedom does it have (where only the incidence structure must be maintained) to 'wiggle' around locally. I understand these are different questions.

I imagine this question would be difficult for an arbitrary configuration. I'm more looking for an upper bound on $M$ that depends on $n,r,b,k$, and perhaps other purely combinatorial parameters.

I've looked at many of the commonly cited sources on the subject of geometric configurations, but I haven't seen anything that relates to this question.

I get a sense that the connectivity of the Levi graph might play some sort of role in bounding this number. I've built simple examples (analyzing the Levi graphs of things like a configuration $C$ with an extra point joined to every vertex, or taking the $k$-fold product of a configuration), but any trends I see are difficult to pin down in general.

Another rather 'trivial' thing I've noticed is that if the configuration is embedded, and we try to move a point $x$ of the configuration a small amount in any direction, then at least $r-1$ lines through $x$ must also move. i.e. if a point is moved, then $r-1$ lines through that point move, and similarly if a line moves from its original position, then $k-1$ points through that line move as well. I've used this idea in the following fashion: suppose we fix three points in an embedded configuration $C$, and let $x$ be another point that is able to be locally moved without destroying the configuration. Then the points/lines that move when $x$ moves form a configuration with replication numbers $r-1,r$ and block sizes $k-1,k$. But the problem from here seems to be that two points moved independently seems to provide nothing stronger (it doesn't look like we can 'predict' what happens in any sense).

I know my exterminations are nothing concrete, and a lot of it is just play, but I'd appreciate any ideas on where to look.

Thanks!

G. Flowers Edit: @Paseman: I have indeed thought about configurations in multiple dimensions projecting onto $\mathbb{R}^2$. However, I don't see much of a relation between a planar realization allowing for $M$ points to be placed in general position, and a configuration that is realizable in $\mathbb{R}^d$ (i.e. a $d-$dimensional configuration). Yes, there is some freedom in how we project, but this seems to provide a lower bound on $M$, rather than an upper one.

I apologize for not being immensely clear. I'm referring to point-line configurations in the Euclidean plane; although, the real projective plane works just as well.

As an example, I would say that $M=5$ for the Pappus Configuration, since five points can be placed in general position, and we can still draw the Pappus configuration with them.

• I may have the wrong idea of where you want to go with this, but maybe this suggestion will apply. Consider embedding the configuration in n-dimensional space for n very high, and get a sense for how the points, lines, and hyperplanes move in this fashion. Unless something weird is going on, you can project these movements down into some two dimensional plane and still get a lot of information on degrees of freedom. Gerhard "That's One Place To Look" Paseman, 2014.01.21 – Gerhard Paseman Jan 21 '14 at 23:32
• @GerhardPaseman: I may have a wrong idea, but can we really embed a configuration to anything but a plane? Once two lines have been chosen, the rest intersects them and hence is bound to lie in the same plane. (Well, I assume that everything is projective.) – Alex Degtyarev Jan 21 '14 at 23:38
• I am unsure about what configurations the poster is talking. My mind suggests the picture of the configuration of Desargues, which has a two and a three dimensional version (which look identical on paper!), and how that can be altered by moving a point while preserving incidence relations. It may be your idea is right; I would like to see an example before making that assumption, even though the poster says there is a planar realization. Besides, there is something that might be revealed in n dimensions before projection. Gerhard "Getting A Really Big Flashlight" Paseman, 2014.01.21 – Gerhard Paseman Jan 21 '14 at 23:48