BSC4024101 Project Details

SupervisorTeo Kok Ming
Project CodeBSC4024101
Title of ProjectSchur's Theorem and its Generalizations
DescriptionSchur's Theorem, proved by Isaai Schur in 1916, is one of the theorems in Ramsey theory on integers.

     Theorem (Schur's Theorem). For every positive integer r, there exists a smallest positive integer s = s(r) such that for every partition {P1,P2,…,Pr} of the set {1,2,…,s} (that is, Pi≠∅ for all 1≤i≤r, r⋃i=1Pi={1,2,…,s} and Pi and Pj are pairwise disjoint for all 1≤i<j≤r), one of the Pi's contains integers x,y,z (need not be distinct) such that x+y=z.

     Let's take r = 2. Note that for the partition {{1,4},{2,3}} of the set {1,2,3,4}, neither {1,4} nor {2,3} contains integers x,y,z such that x+y=z. This shows that s(2)>4. We can verify by brute force that if we partition the set {1,2,3,4,5} into P1 and P2, then one of P1 and P2 must contain integers x,y,z such that x+y=z. For example, for the partition {{1,2},{3,4,5}} of {1,2,3,4,5}, {1,2} contains integers x=y=1 and z=2 with x+y=z. Therefore s(2)=5

     The numbers s(r) in the theorem are known as Schur numbers. To date, the only known values for Schur numbers are s(1)=2, s(2)=5, s(3)=14, s(4)=45, and s(5)=161.

     In this project, we shall study Schur's Theorem and its proof, bounds on Schur numbers, and some generalizations of Schur's Theorem.
Pre-requisitesNo formal pre-requisites.
References[1] Graham, R., Rothschild, B., & Spencer, J. (1990). Ramsey Theory (2nd Ed.). Wiley-Interscience.
[2] Landman, B.M., & Robertson, A. (2015). Ramsey Theory on the Integers (2nd Ed.). American Mathematical Society.