How is this theorem used in applications? I've been searching for it on the web but can't seem to find. Only to "correct codes". Can someone please give a few simple examples?
/lost student
2026-03-25 01:15:04.1774401304
Sylvester-Gallai Theorem
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It might be worth stating the theorem: for any finite set of points in the Euclidean plane either the points are collinear or else there is a line passing through exactly two of the points.
This theorem was probably not proved with any particular applications in mind.
Nevertheless, on my first page of Google results, I find this paper studying higher-dimensional versions of the Sylvester-Gallai theorem. The abstract states that there are applications to error-correcting codes, and in the paper they refer to reader to this paper for more information. It turns out that some generalized, modified version of Sylvester-Gallai allows one to bound the efficiency of locally detectable codes or locally correctable codes: codes where a given codeword, if corrupted somehow, can be recovered by looking at a few other words. I don't know how this works, but the second linked paper might explain it.