Consider a simple, connected graph Γ withnvertices. LetCbe a code of lengthnwith its coordinates corresponding to the vertices of Γ. We defineCas astorage codeon Γ if, for any codewordc∈C, the information at each coordinate ofccan be recovered by accessing its neighboring coordinates. The main problem here is to construct high-rate storage codes on triangle-free graphs. In this paper, we employ the polynomial method to address a question proposed by Barg and Zémor in 2022, demonstrating that the BCH family of storage codes on triangle-free Cayley graphs achieves a unit rate. Furthermore, we generalize the construction of the BCH family and obtain more storage codes of unit rate on triangle-free graphs. We also compare the BCH family with the other known constructions by examining the rate of convergence of 1/(1-R(Cn)) with respect to the lengthn, whereR(Cn) is the rate of codeCn. At last, we reveal a connection between the storage codes on triangle-free graphs and the Ramsey numberR(3,t), which leads to an upper bound for the rate of convergence of 1/(1 -R(Cn)).