Fix l r 2. Let H (r) l+1 be the r-uniform hypergraph obtained from the complete graph K l+1 by enlarging each edge with a set of r − 2 new vertices. Thus H (r) l+1 has (r − 2) l+1 2 + l + 1 vertices and l+1 2 edges. We prove that the maximum number of edges in an n-vertex r-uniform hypergraph containing no copy of H (r) l+1 is (l) r l r n r + o(n r )as n → ∞. This is the first infinite family of irreducible r-uniform hypergraphs for each odd r > 2 whose Turán density is determined. Along the way, we give three proofs of a hypergraph generalization of Turán's theorem. We also prove a stability theorem for hypergraphs, analogous to the Simonovits stability theorem for complete graphs.

A k-path is a hypergraph P k = {e 1 , e 2 , . . . , e k } such that |e i ∩ e j | = 1 if |j − i| = 1 and e i ∩ e j = ∅ otherwise. A k-cycle is a hypergraph C k = {e 1 , e 2 , . . . , e k } obtained from a (k − 1)-path {e 1 , e 2 , . . . , e k−1 } by adding an edge e k that shares one vertex with e 1 , another vertex with e k−1 and is disjoint from the other edges.Let ex r (n, G) be the maximum number of edges in an r-graph with n vertices not containing a given r-graph G. We prove that for fixed r ≥ 3, k ≥ 4 and (k, r) = (4, 3), for large enough n:if k is even and we characterize all the extremal r-graphs. We also solve the case (k, r) = (4, 3), which needs a special treatment. The case k = 3 was settled by Frankl and Füredi.This work is the next step in a long line of research beginning with conjectures of Erdős and Sós from the early 1970's. In particular, we extend the work (and settle a conjecture) of Füredi, Jiang and Seiver who solved this problem for P k when r ≥ 4 and of Füredi and Jiang who solved it for C k when r ≥ 5. They used the delta system method, while we use a novel approach which involves random sampling from the shadow of an r-graph.

For a fixed graph H, we define the rainbow Turán number ex * (n, H) to be the maximum number of edges in a graph on n vertices that has a proper edge-colouring with no rainbow H. Recall that the (ordinary) Turán number ex(n, H) is the maximum number of edges in a graph on n vertices that does not contain a copy of H. For any non-bipartite H we show that ex * (n, H) = (1+o(1))ex(n, H), and if H is colour-critical we show that ex * (n, H) = ex(n, H). When H is the complete bipartite graph K s,t with s ≤ t we show ex * (n, K s,t) = O(n 2−1/s), which matches the known bounds for ex(n, K s,t) up to a constant. We also study the rainbow Turán problem for even cycles, and in particular prove the bound ex * (n, C 6) = O(n 4/3), which is of the correct order of magnitude.

A cancellative hypergraph has no three edges A; B; C with ADBCC: We give a new short proof of an old result of Bolloba´s, which states that the maximum size of a cancellative triple system is achieved by the balanced complete tripartite 3-graph.One of the two forbidden subhypergraphs in a cancellative 3-graph is F 5 ¼ fabc; abd; cdeg: For nX33 we show that the maximum number of triples on n vertices containing no copy of F 5 is also achieved by the balanced complete tripartite 3-graph. This strengthens a theorem of Frankl and Fu¨redi, who proved it for nX3000:For both extremal results, we show that a 3-graph with almost as many edges as the extremal example is approximately tripartite. These stability theorems are analogous to the Simonovits stability theorem for graphs.

No abstract