It is known that for a graph on n vertices n 2 /4 + 1 edges is sufficient for the existence of many triangles. In this paper, we determine the minimum number of edges sufficient for the existence of k triangles intersecting in exactly one common vertex.

Three classes of finite structures are related by extremal properties: complete d-partite d-uniform hypergraphs, d-dimensional affine cubes of integers, and families of 2 d sets forming a d-dimensional Boolean algebra. We review extremal results for each of these classes and derive new ones for Boolean algebras and hypergraphs, several obtained by employing relationships between the three classes. Related partition or coloring problems are also studied for Boolean algebras. Density results are given for Boolean algebras of sets all of whose atoms are the same size. Academic Press

We describe the C 2k+1 -free graphs on n vertices with maximum number of edges. The extremal graphs are unique for n / ∈ {3k − 1, 3k, 4k − 2, 4k − 1}. The value of ex(n, C 2k+1 ) can be read out from the works of Bondy [3], Woodall [14], and Bollobás [1], but here we give a new streamlined proof. The complete determination of the extremal graphs is also new.We obtain that the bound for n 0 (C 2k+1 ) is 4k in the classical theorem of Simonovits, from which the unique extremal graph is the bipartite Turán graph.

For each n and k, we examine bounds on the largest number m so that for any k-coloring of the edges of K n there exists a copy of K m whose edges receive at most k − 1 colors. We show that for k ≥ √ n + Ω(n 1/3 ), the largest value of m is asymptotically equal to the Turán number t(n, n 2 /k ), while for any constant > 0, if k ≤ (1 − ) √ n then the largest m is asymptotically larger than that Turán number.