Given a set of graphs Ψ={G1, G2, …, Gk}, let ex(n; Ψ) denote the greatest size of a graph with order n that contains no subgraph isomorphic to some Gi, 1≤i≤k. One of the main classes of problems in extremal graph theory, known as Tuŕan-type problems, is for given n, Ψ to determine explicitly the function ex (n, Ψ), or to find its asymptotic behavior. Yang Yuansheng investigated the values of ex(n, Ψ) for Ψ={C4} (UTILITAS MATHEMATICA, 41 (1992), 204-210), Garnick investigated them for Ψ={C3, C4} (Journal of Graph Theory, vol.17, no.5 (1993), 633-645) and Alabdullatif investigated them for Ψ={Cn-k+1, …, Cn and Ψ={Pn-k+1,…, Pn}, (1≤k≤n-2) (Bull. Inst. Combin. Appl., 25 (1999)41-52). This paper investigates the values of ex (n, Ψ) forΨ={C3, C4, C5}, n≤42.