A dynamic version of Schur’s problem for rotations on the unit interval

Abstract We introduce a dynamical analog of Schur’s problem by defining the dynamic Schur number S ( k , f ), the smallest integer n such that every k -coloring of a space Y and every orbit segment $$\mathcal {O}_n(x_0,f)$$ O n ( x 0 , f ) under a transformation $$f :Y\rightarrow Y$$ f : Y → Y contains a monochromatic triple $$\{f^a(x_0),f^b(x_0),f^{a+b}(x_0)\}$$ { f a ( x 0 ) , f b ( x 0 ) , f a + b ( x 0 ) } with $$a,b,a+b \le n$$ a , b , a + b ≤ n . This framework extends classical additive Ramsey theory to dynamical systems. We establish general bounds relating S ( k , f ) to the classical Schur numbers and determine some exact values for circle rotations $$T_\alpha (x)=x+\alpha \pmod 1$$ T α ( x ) = x + α ( mod 1 ) under uniform and refined partitions. Computational experiments based on a late acceptance Hill-Climbing heuristic recover all known Schur numbers in the dynamic setting, motivating further exploration through combinatorial dynamics.