Number of ways to select M cohyperplanar points in finite space

[Nick O]

(I don't like the title, since it is a bit misleading. But, I couldn't think of a more descriptive title that fit in the length restrictions.)

A recurring theme in a problem I am exploring is counting the number of subsets of size n in $Z^{d}_{3}$ that have at least m mutually cohyperplanar (dimension d-1) points.

For example, if n=5, m=3, d=2, the question is: "How many ways are there to select 5 points from a 3x3 plane such that any 3 points are mutually collinear?" Because 5 points guarantee a complete line in $Z^{2}_{3}$, the answer is the same as 9 choose 5, specifically 126.

The specific case I am currently interested in, but most likely not the last, is the case with m=7, d=3. So, the question is: "How many ways are there to select n points from $Z^{3}_{3}$ such that any 7 are mutually coplanar?", where n is a variable.

Is it too much to expect that there should be a fairly simple combinatorial answer to this? So far I have not been able to derive it, but I feel that it should exist.

 

[Greg Bernhardt]

I'm sorry you are not generating any responses at the moment. Is there any additional information you can share with us? Any new findings?

 

[Nick O]

Hello, Greg!

No new developments so far. I posted this while I was taking summer classes, and I am working full-time until school starts again next week. I will probably start working on the problem again at that time.

But, I may later be able to post more old information on my approach that may inspire someone else to fill in the blanks.

 

[Incnis Mrsi]

Ī completely misunderstand “5 points guarantee a complete line…” stuff, and preceding question looks bizarre. First of all, in affine space over an arbitrary field any two distinct points unambiguously define a line. Also, in an affine space over ℤ₃ (whichever dimension, 1 or more) a line always consists of exactly 3 points. Three distinct points can be collinear over ℤ₃. More than three distinct points can’t be collinear over ℤ₃ in an affine space.

 

[Nick O]

Thanks for the response.

There are 27 points in Z₃³.

If you choose two of these points at random, you can define a line through those points. However, the set of points you have selected does not contain every point on that line.

You must choose five points to guarantee that the set of points selected contains every point on at least one line.

The question was this: how many points must be selected in order for the set of selected points to contain every point in two parallel lines? The answer is at least six, because it must contain at least the six points in the two lines.