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For graphs \(F\) and \(H\), where \(H\) has chromatic index \(t\), the proper Ramsey number \(PR(F, H)\) is the smallest positive integer \(n\) such that every \(t\)-edge coloring of \(K_n\) results in a monochromatic \(F\) or a properly colored \(H\). The proper Ramsey number \(PR(F, H)\) is investigated for certain pairs \(F, H\) of connected graphs when \(t = 2\), namely when \(F\) is a complete graph, star, or path and when \(H\) is a path or even cycle of small order. In particular, \(PR(F, H)\) is determined when (1) \(F\) is a complete graph and \(H\) is a path of order 6 or less, (2) \(F\) is a complete graph and \(H\) is a 4-cycle, (3) \(F\) is a star and \(H\) is a 4-cycle or a 6-cycle, and (4) \(F\) is a star and \(H\) is a path of order 8 or less.