Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. The Szemeredi–Trotter theorem is a mathematical result in the field of combinatorial geometry. We may discard the lines which contain two or fewer of the points, as they can contribute at most 2m incidences to the total number. Thus we may assume that every line contains at least three of the points. If a line contains k points, then it will contain k?1 line segments which connect two of the n points. In particular it will contain at least k/2 such line segments, since we have assumed k? 3. Adding this up over all of the m lines, we see that the number of line segments obtained in this manner is at least half of the total number of incidences. Thus if we let e be the number of such line segments, it will suffice to show that e = O(n2 / 3m2 / 3 + n + m).

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