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Let H be an r-uniform hypergraph and F be a graph. We say H contains F as a trace if there exists some set S ⊆ V (H) such that H|S := {E ∩ S : E ∈ E(H)} contains a subgraph isomorphic to F. Let exr(n, T r(F )) denote the maximum number of edges of an n-vertex r-uniform hypergraph H which does not contain F as a trace. In this paper, we improve the lower bounds of exr(n, T r(F )) when F is a star, and give some optimal cases. We also improve the upper bound for the case when H is 3-uniform and F is K2,t when t is small.

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Erdős posed the problem of finding conditions on a graph G that imply the largest number of edges in a triangle-free subgraph is equal to the largest number of edges in a bipartite subgraph. We generalize this problem to general cases. Let δ r be the least number so that any graph G on n vertices with minimum degree δ r n has the property P r−1 (G) = K r f (G), where P r−1 (G) is the largest number of edges in an (r − 1)-partite subgraph and K r f (G) is the largest number of edges in a K r -free subgraph. We show that 3r−4 3r−1 < δ r ≤ 4(3r−7)(r−1)+1 4(r−2)(3r−4) when r ≥ 4. In particular, δ 4 ≤ 0.9415.

Grassmannian Gq(n, k) is the set of all k-dimensional subspaces of the vector space F n q . Recently, Etzion and Zhang introduced a new notion called covering Grassmannian code which can be used in network coding solutions for generalized combination networks. An α-(n, k, δ) c q covering Grassmannian code C is a subset of Gq(n, k) such that every set of α codewords of C spans a subspace of dimension at least δ + k in F n q . In this paper, we derive new upper and lower bounds on the size of covering Grassmannian codes. These bounds improve and extend the parameter range of known bounds.