Event Details

Speaker: Tomasz Kościuszko (Adam Mickiewicz University)

Title: Schur-like numbers and a lemma of Shearer

Abstract: Suppose that we colour all integers from 1 to N with n colours. The question of Schur asks about the largest N for which this can be done without creating any monochromatic sums. The exact answer is not known and the factorial bound that Schur proposed has not been improved since 1917. Exactly 100 years later Cwalina and Schoen considered the same problem generalized to other non-invariant equations. In particular they gave a superior bound for an equation in 5 variables. We further improve their result, employing a tool from Graph Theory - Shearer's lemma. In this talk we will sketch the proof and give some motivation as well as discuss several related problems and results.