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

Abstract

It is shown that $r(K_{1,m,k},K_n)\le (k–1+o(1))(\frac{n}{logn}{)}^{m+1}$ for any two fixed integers $k\ge m\ge 2$ and $n\to \mathrm{\infty }$.
This result is obtained using the analytic method and the function $f_m(x)={\int }_0^1\frac{(1-t{)}^{\frac{1}{m}}dt}{m+(x-m{)}^t},x\ge 0,m\ge 1,$
building upon the upper bounds for $r(K_{m,k},K_n)$ established by Y. Li and W. Zang.Furthermore, $(c–o(1))(\frac{n}{logn}{)}^{\frac{7}{3}}\le r(W_4,K_n)\le (1+o(1))(\frac{n}{logn}{)}^3$ (as $n\to \mathrm{\infty }$). Moreover, we derive
$r(K_1+K_{m,k},K_n)\le (k–1+o(1))(\frac{n}{logn}{)}^{l+m}$ for any two fixed integers $k\ge m\ge 2$ (as $n\to \mathrm{\infty }$).