Asymptotic Upper Bounds for \(K_{1,m,k}\): Complete Graph Ramsey Numbers

Abstract

It is shown that \(r(K_{1,m,k}, K_n) \leq (k – 1 + o(1)) (\frac{n}{log n})^{m+1}\) for any two fixed integers \(k \geq m \geq 2\) and \(n \to \infty\).
This result is obtained using the analytic method and the function \(f_{m}(x) = \int_0^1 \frac{(1-t)^{\frac{1}{m}}dt}{m+(x-m)^t} , \quad x \geq 0,m \geq 1,\)
building upon the upper bounds for \(r(K_{m,k}, K_n)\) established by Y. Li and W. Zang.Furthermore, \((c – o(1)) (\frac{n}{log n})^{\frac{7}{3}}\leq r(W_{4}, K_n) \leq (1 + o(1)) (\frac{n}{log n})^{3}\) (as \(n \to \infty\)). Moreover, we derive
\(r(K_{1} + K_{m,k}, K_n) \leq (k – 1 + o(1)) (\frac{n}{log n})^{l+m}\) for any two fixed integers \(k \geq m \geq 2\) (as \(n \to \infty\)).