Graduate Program in Mathematics.Demir, Selçuk.Budak, Talin.Şimşek, Sevim.2023-03-162023-03-162009.MATH 2009 S56https://digitalarchive.library.bogazici.edu.tr/handle/123456789/15241In this thesis we first present two examples from Geometric Ramsey theory in R2. Then we generalize these results to higher dimensions and construct our main theorem. Then we translate our geometric problem into dynamical form. Finally we prove the main theorem by using methods from ergodic theory.30cm.Ramsey theory.Ergodic theory.viii, 43 leaves;