  We study the online graph coloring problem restricted to the intersection
graphs of intervals with lengths in $[1,\sigma]$. For $\sigma=1$ it is the
class of unit interval graphs, and for $\sigma=\infty$ the class of all
interval graphs. Our focus is on intermediary classes.
  We present a $(1+\sigma)$-competitive algorithm, which beats the state of the
art for $1 < \sigma < 2$, and proves that the problem we study can be strictly
easier than online coloring of general interval graphs.
  On the lower bound side, we prove that no algorithm is better than
$5/3$-competitive for any $\sigma>1$, nor better than $7/4$-competitive for any
$\sigma>2$, and that no algorithm beats the $5/2$ asymptotic competitive ratio
for all, arbitrarily large, values of $\sigma$. That last result shows that the
problem we study can be strictly harder than unit interval coloring. Our main
technical contribution is a recursive composition of strategies, which seems
essential to prove any lower bound higher than $2$.
