| Abstract [eng] |
This master's thesis explores two main topics: a version of the Erdős–Turán problem in groups Zn and Sperner-type theorems for multisets. The first part of this thesis is concerned with explaining the motivation behind this version of the Erdős–Turán problem and the proof that for sufficiently large n there always exists an additive base of the group Zn with representation function bounded above by Clog(n). We also prove that for n large enough one can find an additive basis of order k>2 of Zn with representation function values not exceeding C_klog(n) on average. The second part of this thesis contains an elementary proof of Sperner's theorem using only mathematical induction, and a new proof of a similar result for families of multisets with bounded element multiplicities. |
