Timeline for Szemeredi-Trotter bounds when the lines are implicitly described by a point set
Current License: CC BY-SA 3.0
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| Feb 7, 2018 at 17:57 | comment | added | GMB | @Seva Thanks for the responses -- I've edited to clarify the question. The misunderstanding is that I'm asking about the incidences between the lines determined by these $s$ points (which are in general position w.r.t. each other) and a separate set of $n$ points (which can be in whatever position you like). | |
| Feb 7, 2018 at 17:56 | history | edited | GMB | CC BY-SA 3.0 |
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| Feb 7, 2018 at 17:06 | comment | added | Seva | And, BTW, what do you mean by "$s$ points in general position"? This normally means that no line contains three (or more) points. In this case there are exactly two incidences per each of the $\binom s2$ lines, so that there are $2l$ incidences. | |
| Feb 7, 2018 at 17:05 | comment | added | Sean English | I think I am misunderstanding your question. It seems to me that if you have $s$ points in general position and consider all lines between them, you will always have exactly $2\binom{s}2$ point-line incidences, exactly two for each line. Is there something I am missing here? | |
| Feb 7, 2018 at 16:58 | comment | added | Seva | That is, you want to improve the estimate in the special case where the set of lines includes all lines determined by our point set, not just some "very reach" lines? | |
| Feb 7, 2018 at 16:51 | comment | added | GMB | @Seva the theorem is certainly true, but the corresponding lower bound that implies optimality of the Szemeredi-Trotter Theorem no longer holds. I am wondering if an asymptotically smaller upper bound can be obtained in this restricted setting. | |
| Feb 7, 2018 at 16:46 | comment | added | Seva | Exactly what would you like to know? The theorem, of course, is perfectly true in your special case, too. | |
| Feb 7, 2018 at 15:58 | history | asked | GMB | CC BY-SA 3.0 |