Extremal Frobenius numbers in a class of sets

Abstract.

For given $A_k=\{ a_1,\ldots ,a_k \}, a_1 \le \ldots \le a_k$ coprime the Frobenius number ${g}(A_k)$ is defined as the greatest integer ${g}$ with no representation¶¶${g}=\sum \limits ^k_{i=1}\,x_i\,a_i,\;x_i\in {\Bbb N}_0$. ¶¶A class ${\bf A}^*_k$ is given, such that ¶¶${\overline {g}}^*(k,y):= \max \{ {g}(A_k)|A_k\in {\bf A}^*_k,\, a_k\le y \}$¶¶has the same asymptotic behaviour as the general function¶¶${\overline {g}}(k,y):= \max \{ {g}(A_k)| a_k\le y \}\, {\rm for} \, y\to \infty$.¶¶ Furthermore, ¶¶${\underline {g}}^*(k,x):= \min \{ {g}(A_k)|A_k\in {\bf A}^*_k,\, a_1\ge x \}$¶¶is shown to have the same order of magnitude as the general function¶¶${\underline {g}}(k,x):= \min \{ {g}(A_k)| a_1\ge x \}\,{\rm for} \, x\to \infty$.