We use numerical bootstrap techniques to study correlation functions of a traceless symmetric tensors of O(N) with two indexes t_{ij}. We obtain upper bounds on operator dimensions for all the relevant representations and several values of N. We discover several families of kinks, which do not correspond to any known model and we discuss possible candidates. We then specialize to the case N=4, which has been conjectured to describe a phase transition in the antiferromagnetic real projective model ARP^{3}. Lattice simulations provide strong evidence for the existence of a second order phase transition, while an effective field theory approach does not predict any fixed point. We identify a set of assumptions that constrain operator dimensions to a closed region overlapping with the lattice prediction. The region is still present after pushing the numerics in the single correlator case or when considering a mixed system involving t and the lowest dimension scalar singlet.
