This project on exploring the infinite is mainly devoted to the theoretical investigation of the recently discovered phenomenon of phase transitions in logic and combinatorics, and to applying these new methods to computer science and to long-standing open problems in mathematics. The goal of PTLC (Phase Transitions in Logic and Combinatorics) is to explore this new phenomenon and to apply the new methods to logic, computer science, and to several open problems in Mathematics that could not be cracked by their traditional communities where progress on them has now been stuck for decades. PTLC will also explore the relationship between phase transitions for independence results and phase transitions in probalistic settings and statistical physics. An additional PTLC-research topic is to study the influence of the infinite on describing certain very large natural numbers and related phase transitions.