  We have investigated, using effective mass approach (EMA), magnetic
properties of a one-dimensional electron gas in graphene armchair ribbons when
the electrons of occupy only the lowest conduction subband. We find that
magnetic properties of the one-dimensional electron gas may depend sensitively
on the width of the ribbon. For ribbon widths $L_x=3Ma_0$, a critical point
separates ferromagnetic and paramagnetic states while for $L_x=(3M+1)a_0$
paramagnetic state is stable ($M$ is an integer and $a_{0}$ is the length of
the unit cell). These width-dependent properties are a consequence of
eigenstates that have a subtle width-dependent mixture of $\mathbf{K}$ and
$\mathbf{K'}$ states, and can be understood by examining the wavefunction
overlap that appears in the expression for the many-body exchange self-energy.
Ferromagnetic and paramagnetic states may be used for spintronic purposes.
