Abstract. Let [Formula: see text] be the book graph which consists of [Formula: see text] copies of triangles all sharing a common edge. Let [Formula: see text] be a cycle of length [Formula: see text]. In 1978, Rousseau and Sheehan initiated the study of the book–cycle Ramsey number. A lot of effort has been made to determine the value of [Formula: see text] since then. In [ Ars Combin., 31 (1991), pp. 239–248], Faudree, Rousseau, and Sheehan mentioned the following: “we know practically nothing about [Formula: see text] when [Formula: see text] is even and greater than four. Also, the problem of computing [Formula: see text] when [Formula: see text] is odd and [Formula: see text] and [Formula: see text] are nearly equal provides an unanswered test of strength.” Answering the second part of the question above, the second and fifth authors recently obtained the value of [Formula: see text] for [Formula: see text] and [Formula: see text] being large. However, the value of [Formula: see text] is previously unknown for [Formula: see text] and [Formula: see text] being even as well as [Formula: see text] and [Formula: see text] being odd. In this paper, for even [Formula: see text], we manage to determine the value of [Formula: see text] provided that [Formula: see text] is linear with [Formula: see text] and [Formula: see text] is large enough. Thus this makes progress towards the first part of the question above. In addition, for odd [Formula: see text], we are able to obtain the value of [Formula: see text] for [Formula: see text] and [Formula: see text] being large.