Turán-type, colouring and matching questions for hypergraphs, where far less is known than for graphs.
Let f(n;r,k) be the maximal number of edges in an r-uniform hypergraph which contains no set of k many independent edges. For all r≥ 3, f(n;r,k)=max(C(rk-1, r), C(n, r)-C(n-k+1, r)). Note: the source states the formula with no range on n or k, but some restriction is needed: e.g.
Let R_r(n) denote the r-uniform hypergraph Ramsey number: the minimal m such that if we 2-colour all edges of the complete r-uniform hypergraph on m vertices then there must be some monochromatic copy of the complete r-uniform hypergraph on n vertices.
Let F(n,α) denote the smallest m such that there exists a 2-colouring of the edges of K_n so that every X⊆ [n] with lvert Xrvert≥ m contains more than α C(lvert Xrvert, 2) many edges of each colour. Prove that, for every 0≤ α < 1/2, F(n,α)∼ c_αlog n for some constant c_α depending only on α.
Let R_3(n) be the minimal m such that if the edges of the 3-uniform hypergraph on m vertices are 2-coloured then there is a monochromatic copy of the complete 3-uniform hypergraph on n vertices. Is there some constant c>0 such that R_3(n) ≥ 2^2^cn?
Erdős Problem 593 (\500): Characterize those finite 3-uniform hypergraphs which appear in every 3-uniform hypergraph of chromatic number > aleph_0. The answer is the set of obligatory finite 3-uniform hypergraphs, represented here on the labelled vertex sets Fin n.
Does there exist a k>2 such that the k-sized subsets of 1,...,2k can be coloured with k+1 colours such that for every A⊂ 1,…,2k with lvert Arvert=k+1 all k+1 colours appear among the k-sized subsets of A?
