  We apply bootstrap techniques in order to constrain the CFT data of the
$(A_1,A_2)$ Argyres-Douglas theory, which is arguably the simplest of the
Argyres-Douglas models. We study the four-point function of its single Coulomb
branch chiral ring generator and put numerical bounds on the low-lying spectrum
of the theory. Of particular interest is an infinite family of semi-short
multiplets labeled by the spin $\ell$. Although the conformal dimensions of
these multiplets are protected, their three-point functions are not. Using the
numerical bootstrap we impose rigorous upper and lower bounds on their values
for spins up to $\ell=20$. Through a recently obtained inversion formula, we
also estimate them for sufficiently large $\ell$, and the comparison of both
approaches shows consistent results. We also give a rigorous numerical range
for the OPE coefficient of the next operator in the chiral ring, and estimates
for the dimension of the first R-symmetry neutral non-protected multiplet for
small spin.
