An explicit construction of (3,t)(3,t)-existentially closed graphs

Let n,tn,t be positive integers. A tt-edge-colored graph GG is (n,t)(n,t)-e.c. or (n,t)(n,t)-existentially closed if for any tt disjoint sets of vertices A1,…,AtA1,…,At with |A1|+⋯+|At|=n|A1|+⋯+|At|=n, there is a vertex xx not in A1∪⋯∪AtA1∪⋯∪At such that all edges from this vertex to the set AiAi are colored by the ii-th color. In this paper, we give an explicit construction of a (3,t)(3,t)-e.c. graph of polynomial order.