For fixed k, r̃(P_k, P_n)/n and r̃(P_k, C_n)/n converge to λ1 while r̃(K_{1,k}, P_n)/n and r̃(K_{1,k}, C_n)/n converge to λ2 as n tends to infinity.
G., and Erd ˝os, P.Some linear and some quadratic recursion formulas
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On the asymptotic behavior of online Ramsey numbers for stars, paths and cycles
For fixed k, r̃(P_k, P_n)/n and r̃(P_k, C_n)/n converge to λ1 while r̃(K_{1,k}, P_n)/n and r̃(K_{1,k}, C_n)/n converge to λ2 as n tends to infinity.