A structural result for hypergraphs with many restricted edge colorings

Hanno Lefmann, Yury Person, Mathias Schacht
2010 Journal of Combinatorics  
In honor of Joel Spencer's 60+ birthday For k-uniform hypergraphs F and H and an integer r ≥ 2, let c r,F (H) denote the number of r-colorings of the set of hyperedges of H with no monochromatic copy of F and let c r,F (n) = max H∈Hn c r,F (H), where the maximum runs over the family H n of all k-uniform hypergraphs on n vertices. Moreover, let ex(n, F ) be the usual Turán function, i.e., the maximum number of hyperedges of an n-vertex k-uniform hypergraph which contains no copy of F . In this
more » ... py of F . In this paper, we consider the question for determining c r,F (n) for arbitrary fixed hypergraphs F and show c r,F (n) = r ex(n,F )+o(n k ) for r = 2, 3. Moreover, we obtain a structural result for r = 2, 3 and any H with c r,F (H) ≥ r ex(|V (H)|,F ) under the assumption that a stability result for the k-uniform hypergraph F exists and |V (H)| is sufficiently large. We also obtain exact results for c r,F (n) when F is a 3-or 4-uniform generalized triangle and r = 2, 3, while c r,F (n) r ex(n,F ) for r ≥ 4 and n sufficiently large.
doi:10.4310/joc.2010.v1.n4.a4 fatcat:s3lyjqrrvvcpxhgodol6fprpae