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A Constraint Satisfaction Problem (CSP) is a computational problem where weare given variables and constraints about them; the question is whether thevariables can be assigned values such that all constraints are satisfied. Wegive an overview of the current state of research on CSPs where values for thevariables and constraints are taken from a finitely bounded homogeneousstructure which is fixed beforehand. We explain the main mathematical ideas sofar, the three dilemmas they brought upon us, and what could be done toovercome them in order to obtain a satisfactory understanding of thecomputational complexity of such CSPs.

Current Challenges in Infinite-Domain Constraint Satisfaction: Dilemmas of the Infinite Sheep