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, ,q\ { 0 , 1 , ⋯ , q } and every vector (that we call a q -edge) having exactly two non-zero entries. The support of a q -edge x x is the pair Sₓ S x of indices of non-zero entries. We say that H is an s -copy of an ordinary graph F if $$|H|=|E(F)|$$ | H | = | E ( F ) | , F is isomorphic to the graph with edge set ₓ:x∈ H\ { S x : x ∈ H } , and whenever v∈ e,e'∈ E(F) v ∈ e , e ′ ∈ E ( F ) , the entries with index corresponding to v in the q -edges corresponding to e and $$e'$$ 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 \!ex(n,F,q,s) 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 \!ex(n,F,q,q+1) ex ( n , F , q , q + 1 ) for many graphs F .