Independence number of intersection graphs of axis-parallel segments

We prove that for any triangle-free intersection graph of $n$ axis-parallel segments in the plane, the independence number $\alpha$ of this graph is at least $\alpha \ge n\mathrm{/}4+\mathrm{\Omega }(\sqrt{n})$. We complement this with a construction of a graph in this class satisfying $\alpha \le n\mathrm{/}4+c\sqrt{n}$ for an absolute constant $c$, which demonstrates the optimality of our result.