The main result of this paper is a bound on the quantity $A$ that is related to the 1-loop Beta function in scalar theories with quartic interactions in the 4-$\epsilon$ expansion, whose symmetry group is always a subgroup of $O(N)$. A theorem of Michel shows that this $A$ is related to the stability of RG flows at this order, such that fixed points with minimal $A$ ($A$ is negative) are stable to this order.
The authors verify that the $O(N)$ fixed points for $1\leq N\leq 4$, which are known to be 1-loop stable, indeed have the minimal values $A$ of all theories that they checked, and the $O(4)$ theory in fact saturates the bound. This bound on $A$ is especially interesting for $ N>2$, where such scalar theories have not yet been classified. For $N>4$, they find that other scalar theories, such as the tetrahedral theory for $N=5$ and the bifundamental theory for $N=511$, saturate the bound.
Secondary results include:
1. Derivation that the quartic potential must be positive for fixed points and that QFTs with negative potentials cannot flow to fixed points
2. Resolution of confusion regarding zero eigenvalues of the linearized RG equations.
The paper also includes a review of known results of scalar theories with quartic potentials, collecting results in the literature that might not be known to many readers.
In the conclusion, the authors mention that similar results at 1-loop also apply to theories with fermions. To test the validity of these 1-loop results non-perturbatively, it would be useful to consider supersymmetric theories in which non-perturbative results are available.