For　graphs　H1　and　H2,　the　Ramsey　number　r(H1,H2)　is　the　smallest　positive　integer　N　such　that　any　graph　G　on　N　vertices　contains　H1　as　a　subgraph,　or　its　complement　contains　H2　as　a　subgraph.　Let　Bn(k)　denote　the　book　graph　on　n+k　vertices　which　consists　of　n　copies　of　Kk+1　all　sharing　a　common　Kk,　and　let　H:=Kp(a1,⋯,ap)　be　the　complete　p-partite　graph　with　parts　of　sizes　a1,⋯,ap.　Recently,　strengthening　a　result　of　Fox,　He　and　Wigderson　(Adv.　Combin.　4　(2023),　21pp),　Fan　and　Lin　(J.　Combin.　Theory　Ser.　A　199　(2023),　19pp)　showed　that　for　every　k,p,t≥2,　there　exists　δ＞0　such　that　the　following　holds　for　all　large　n.　Let　1≤a1≤⋯≤ap-1≤t　and　ap≤δn　be　positive　integers.　If　a1=1,　then　r(H,Bn(k))≤(p-1)(n+ka2-1)+1.　The　inequality　is　tight　if　n≡1(moda2).　In　this　paper,　we　improve　the　above　upper　bounds　for　the　cases　when　n≡2(moda2)　and　n≡3(moda2).　Combining　the　new　upper　bounds　and　constructions　of　the　lower　bounds　for　these　cases,　we　are　able　to　determine　the　exact　values　of　r(Kp(a1,⋯,ap),Bn(k))　when　p=3.　The　bound　on　1/δ　we　obtain　is　not　of　tower-type　since　our　proof　does　not　rely　on　the　regularity　lemma.　©　The　Author(s),　under　exclusive　licence　to　Springer　Nature　Japan　KK　2024.