Appears in collection : 2018 - T1 - WS3 - Model theory and applications

I will mention some combinatorial problems related to dense structures. For example, if n points in the plane determine cn2 lines with at least three 6 points on them, then one expects that many of the points are in a special position. Potential tools to analyse such arrangements are Regularity Lemmas and combinatorial variants of the Group Configuration Theorem. Unfortunately even these powerful techniques are not enough to answer some simple questions. For example in the point-line arrangement described above, one would expect that if n is large enough then some (say 10) points are on an elliptic curve, but this problem seems to be out of reach.