Schur numbers involving rainbow colorings

Gallai–Ramsey Numbers for Rainbow Paths

“…Due to the rapid development of the Gallai-Ramsey number in the past decade, it has become a hot research area in graph theory. Inspired by this problem, the definition of the Gallai-Schur number was proposed in Budden's paper [3] in 2020, introducing this type of problem from graph theory to number theory. Since the Gallai-Schur number only studies the equation x + y = z, it can be generalized to other equations, and similarly defined as the Gallai-Rado number.…”

Anti-van der Waerden Numbers on Graphs

“…Additionally, rainbow solutions to linear equations in [n] were studied in [6] and [9]. Interestingly, the authors in [6] and [9] produced similar results simultaneously but using differing arguments. The definition of aw([n], k) was also extended to Z n to get aw(Z n , k) in [7] which led to results on arithmetic progression in finite abelian groups by Young in [20].…”

“…It should also be noted that aw([n], 3) was studied in [7] and the exact value was determined in [4]. Additionally, rainbow solutions to linear equations in [n] were studied in [6] and [9]. Interestingly, the authors in [6] and [9] produced similar results simultaneously but using differing arguments.…”