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.

In this paper, we study $r$-uniform hypergraphs ${\cal H}$ without cycles of length less than five, employing the definition of a hypergraph cycle due to Berge. In particular, for $r = 3$, we show that if ${\cal H}$ has $n$ vertices and a maximum number of edges, then $$|{\cal H}|={\textstyle 1\over6}n^{3/2} + o(n^{3/2}).$$ This also asymptotically determines the generalized Turán number $T_{3}(n,8,4)$. Some results are based on our bounds for the maximum size of Sidon-type sets in $\Bbb{Z}_{n}$.

A recently posed question of Häggkvist and Scott's asked whether or not there exists a constant c such that if G is a graph of minimum degree ck then G contains cycles of k consecutive even lengths. In this paper we answer the question by proving that for k ≥ 2, a bipartite graph of average degree at least 4k and girth g contains cycles of (g/2 − 1)k consecutive even lengths. We also obtain a short proof of the theorem of Bondy and Simonovits, that a graph of order n and size at least 8(k − 1)n 1+1/k has a cycle of length 2k.Erdős and Burr [4] conjectured that for every odd number k, there is a constant c k such that for every natural number m, every graph of average degree at least c k contains a cycle of length m modulo k. Erdős and Burr [4] settled their conjecture in the case m = 2 and Robertson (see [4]) settled the case m = 0. The full conjecture was resolved by Bollobás [1], who proved the conjecture with c k = 2[(k + 1) k − 1]/k. In this paper, we show that c k = 8k will do. Thomassen [11] later showed cycles of all even lengths modulo k are obtained under the hypothesis that the average degree is at least 4k(k + 1), without requiring k to be odd. Thomassen [10] also proved that if G is a graph of minimum degree at least three and girth at least 2(k 2 + 1)(3 · 2 k 2 +1 + (k 2 + 1) 2 − 1), then G contains cycles of all even lengths modulo k.

For an odd integer k, let C k = {C 3 , C 5 , . . . , C k } denote the family of all odd cycles of length at most k and let C denote the family of all odd cycles. Erdős and Simonovits [10] conjectured that for every family F of bipartite graphs, there exists k such that ex n, F ∪ C k ∼ ex n, F ∪ C as n → ∞. This conjecture was proved by Erdős and Simonovits when F = {C 4 }, and for certain families of even cycles in [13]. In this paper, we give a general approach to the conjecture using Scott's sparse regularity lemma. Our approach proves the conjecture for complete bipartite graphs K 2,t and K 3,3 : we obtain more strongly that for any odd k ≥ 5, ex n, F ∪ {C k } ∼ ex n, F ∪ C and we show further that the extremal graphs can be made bipartite by deleting very few edges. In contrast, this formula does not extend to triangles -the case k = 3 -and we give an algebraic construction for odd t ≥ 3 of K 2,t -free C 3 -free graphs with substantially more edges than an extremal K 2,t -free bipartite graph on n vertices. Our general approach to the Erdős-Simonovits conjecture is effective based on some reasonable assumptions on the maximum number of edges in an m by n bipartite F -free graph.