The well-known Sylvester–Gallai Theorem states that a set of $n>2$ points in $R^2$ not all on a line contains two points such that the line passing through these two points does not contain a third point in the set.

One of the reasons this is somewhat tricky to prove is that it is false in some geometries, including finite field geometry. This can be seen by considering, for example, all $p^2$ points in $F_p^2$ a plane over a finite field.

My question is the following:

Let $n$ be a set of (say) $n=p^{1/100}$ points in $F_p^2$ a large finite field of prime order. Then does the conclusion of the Sylvester-Gallai theorem hold?

I am restricting the question to prime order fields to (1) avoid issues with subfields, and to (2) try to prevent the analog of the Hesse Configuration, a counterexample to the corresponding statement over the complex numbers.

Many of the proofs of the real case use the ordering of the field, so will run into trouble in a finite field. On the other hand, once the sets are really really small (much smaller than what is being considered here) there might be hope of formally translating the problem to the complexes and using the characterizations of possible counter-examples [see: https://arxiv.org/pdf/math/0403023.pdf].

I assume this question would been studied before, but I can't find a reference. I'm mostly interested in a positive result, so if there is a counter-example, I'd be interested to know if there is an understanding of counterexamples, which there is in the complex case.

The well-known Sylvester–Gallai Theorem states that a set of $n>2$ points in $R^2$ not all on a line contains two points such that the line passing through these two points does not contain a third point in the set.

One of the reasons this is somewhat tricky to prove is that it is false in some geometries, including finite field geometry. This can be seen by considering, for example, all $p^2$ points in $F_p^2$ a plane over a finite field.

My question is the following:

Let $n$ be a set of (say) $n=p^{1/100}$ points in $F_p^2$ a large finite field of prime order. Then does the conclusion of the Sylvester-Gallai theorem hold?

I am restricting the question to prime order fields to (1) avoid issues with subfields, and to (2) try to prevent the analog of the Hesse Configuration, a counterexample to the corresponding statement over the complex numbers.

Many of the proofs of the real case use the ordering of the field, so will run into trouble in a finite field. On the other hand, once the sets are really really small (much smaller than what is being considered here) there might be hope of formally translating the problem to the complexes and using the characterizations of possible counter-examples [see: https://arxiv.org/pdf/math/0403023.pdf].

I assume this question would been studied before, but I can't find a reference. I'm mostly interested in a positive result, so if there is a counter-example, I'd be interested to know if there is an understanding of counterexamples, which there is in the complex case.

The well-known Sylvester–Gallai Theorem states that a set of $n>2$ points in $R^2$ not all on a line contains two points such that the line passing through these two points does not contain a third point in the set.

One of the reasons this is somewhat tricky to prove is that it is false in some geometries, including finite field geometry. This can be seen by considering, for example, all $p^2$ points in $F_p^2$ a plane over a finite field.

My question is the following:

Let $n$ be a set of (say) $n=p^{1/100}$ points in $F_p^2$ a large finite field of prime order. Then does the conclusion of the Sylvester-Gallai theorem hold?

I am restricting the question to prime order fields to (1) avoid issues with subfields, and to (2) try to prevent the analog of the Hesse Configuration, a counterexample to the corresponding statement over the complex numbers. Many of the proofs of the real case use the ordering of the field, so will run into trouble in a finite field.

  On the other hand, once the sets are really really small (much smaller than what is being considered here) there might be hope of formally translating the problem to the complexes and using the characterizations of possible counter-examples [see: https://arxiv.org/pdf/math/0403023.pdf].

I assume this question would been studied before, but I can't find a reference.

The well-known Sylvester–Gallai Theorem states that a set of $n>2$ points in $R^2$ not all on a line contains two points such that the line passing through these two points does not contain a third point in the set.

One of the reasons this is somewhat tricky to prove is that it is false in some geometries, including finite field geometry. This can be seen by considering, for example, all $p^2$ points in $F_p^2$ a plane over a finite field.

My question is the following:

Let $n$ be a set of (say) $n=p^{1/100}$ points in $F_p^2$ a large finite field of prime order. Then does the conclusion of the Sylvester-Gallai theorem hold?

I am restricting the question to prime order fields to (1) avoid issues with subfields, and to (2) try to prevent the analog of the Hesse Configuration, a counterexample to the corresponding statement over the complex numbers. 

Many of the proofs of the real case use the ordering of the field, so will run into trouble in a finite field. On the other hand, once the sets are really really small (much smaller than what is being considered here) there might be hope of formally translating the problem to the complexes and using the characterizations of possible counter-examples [see: https://arxiv.org/pdf/math/0403023.pdf].

I assume this question would been studied before, but I can't find a reference. I'm mostly interested in a positive result, so if there is a counter-example, I'd be interested to know if there is an understanding of counterexamples, which there is in the complex case.