Extremal Graph Theoretic Questions for q-Ary Vectors

A q-graph H on n vertices is a set of vectors of length n with all entries from \(\{0,1,\dots ,q\}\) and every vector (that we call a q-edge) having exactly two non-zero entries. The support of a q-edge \({\textbf{x}}\) is the pair \(S_{\textbf{x}}\) of indices of non-zero entries. We say that H is an s-copy of an ordinary graph F if \(|H|=|E(F)|\), F is isomorphic to the graph with edge set \(\{S_{\textbf{x}}:{\textbf{x}}\in H\}\), and whenever \(v\in e,e'\in E(F)\), the entries with index corresponding to v in the q-edges corresponding to e and \(e'\) sum up to at least s. E.g., the q-edges (1, 3, 0, 0, 0), (0, 1, 0, 0, 3), and (3, 0, 0, 0, 1) form a 4-triangle. The Turán number \(\mathop {}\!\textrm{ex}(n,F,q,s)\) is the maximum number of q-edges that a q-graph H on n vertices can have if it does not contain any s-copies of F. In the present paper, we determine the asymptotics of \(\mathop {}\!\textrm{ex}(n,F,q,q+1)\) for many graphs F.