Recently, it became apparent that a large number of the most interesting structures and
phenomena of the world can be described by networks. To develop a mathematical theory of …

Using the formalism of flag algebras, we prove that every triangle-free graph G with n
vertices contains at most (n/5) 5 cycles of length five. Moreover, the equality is attained only …

Every 3-graph in which no four vertices are independent and no four vertices span precisely
three edges must have edge density \geq4/9(1-o(1)). This bound is tight. The proof is a …

We survey some recent results on long-standing conjectures regarding Hamilton cycles in
directed graphs, oriented graphs and tournaments. We also combine some of these to prove …

A graph H is called common if the sum of the number of copies of H in a graph G and the
number in the complement of G is asymptotically minimized by taking G to be a random …

Let C (n) denote the maximum number of induced copies of 5-cycles in graphs on n vertices.
For n large enough, we show that C (n)= a⋅ b⋅ c⋅ d⋅ e+ C (a)+ C (b)+ C (c)+ C (d)+ C (e) …

In this paper we modify slightly Razborov's flag algebra machinery to be suitable for the
hypercube. We use this modified method to show that the maximum number of edges of a 4 …

Boros and Füredi (for d= 2) and Bárány (for arbitrary d) proved that there exists a positive
real number cd such that for every set P of n points in R d in general position, there exists a …

A famous conjecture of Caccetta and Häggkvist is that in a digraph on vertices and minimum
outdegree at least n/r there is a directed cycle of length or less. We consider the following …

We consider the problem of determining the maximum induced density of a graph H in any
graph on n vertices. The limit of this density as n tends to infinity is called the inducibility of H …