 First Dahlquist barrier

For a consistent zero-stable real linear $k$-step method with constant coefficients and only first-derivative evaluations, the attainable order is at most $k+1$ for odd $k$ and $k+2$ for even $k$. For an explicit method the bound is $k$. These are order barriers under the root condition for a multistep method, not bounds on multiderivative methods or on arbitrary variable-step formulations. The Second Dahlquist barrier adds the stronger bound of order two for irreducible A-stable methods.