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We take a long magical tour in algebraic logic, starting from classicalresults on neat embeddings due to Henkin, Monk and Tarski, all the way torecent results in algebraic logic using so--called rainbow constructionsinvented by Hirsch and Hodkinson. Highlighting the connections with graphtheory, model theory, and finite combinatorics, this article aspires to presenttopics of broad interest in a way that is hopefully accessible to a largeaudience. The paper has a survey character but it contains new approaches toold ones. We aspire to make our survey fairly comprehensive, at least in so faras Tarskian algebraic logic, specifically, the theory of cylindric algebras, isconcerned. Other topics, such as abstract algebraic logic, modal logic and theso--called (central) finitizability problem in algebraic logic will be dealtwith; the last in some detail. Rainbow constructions are used to solve problemsadressing classes of cylindric--like algebras consisting of algebras having aneat embedding property. The hitherto obtained results generalize seminalresults of Hirsch and Hodkinson on non--atom canonicity, non--first orderdefinabiity and non--finite axiomatizability, proved for classes ofrepresentable cylindric algebras of finite dimension$>2$. We show that suchresults remain valid for cylindric algebras possesing relativized {\it cliqueguarded} representations that are {\it only locally} well behaved. The paper iswritten in a way that makes it accessible to non--specialists curious about thestate of the art in Tarskian algebraic logic. Reaching the boundaries ofcurrent research, the paper also aspires to be informative to the practitioner,and even more, stimulates her/him to carry on further research in main streamalgebraic logic.

A brief history of algebraic logic from neat embeddings to rainbow constructions