155 research outputs found
A hierarchy of randomness for graphs
AbstractIn this paper we formulate four families of problems with which we aim at distinguishing different levels of randomness.The first one is completely non-random, being the ordinary Ramsey–Turán problem and in the subsequent three problems we formulate some randomized variations of it. As we will show, these four levels form a hierarchy. In a continuation of this paper we shall prove some further theorems and discuss some further, related problems
On the Ramsey-Tur\'an density of triangles
One of the oldest results in modern graph theory, due to Mantel, asserts that
every triangle-free graphs on vertices has at most
edges. About half a century later Andr\'asfai studied dense triangle-free
graphs and proved that the largest triangle-free graphs on vertices without
independent sets of size , where , are blow-ups
of the pentagon. More than 50 further years have elapsed since Andr\'asfai's
work. In this article we make the next step towards understanding the structure
of dense triangle-free graphs without large independent sets.
Notably, we determine the maximum size of triangle-free graphs~ on
vertices with and state a conjecture on the structure of
the densest triangle-free graphs with . We remark that the
case behaves differently, but due to the work of Brandt
this situation is fairly well understood.Comment: Revised according to referee report
Phase transitions in the Ramsey-Turán theory
Let f(n) be a function and L be a graph. Denote by RT(n, L, f(n)) the maximum number of edges of an L-free graph on n vertices with independence number less than f(n). Erdos and Sós asked if RT (n, K5, c√
n) = o (n2) for some constant c. We answer this question by proving the stronger RT(n, K5, o (√n log n)) = o(n2). It is known that RT (n, K5, c√n log n
)= n2/4 + o (n2) for c > 1, so one can say that K5 has a Ramsey-Turán-phase transition at c√n log n. We extend this result to several other Kp's and functions f(n), determining many more phase transitions. We shall formulate
several open problems, in particular, whether variants of the Bollobás-Erdos graph, which is a geometric construction, exist to give good lower bounds
on RT (n, Kp, f(n)) for various pairs of p and f(n). These problems are studied in depth by Balogh-HuSimonovits, where among others, the Szemerédi's Regularity Lemma and the Hypergraph Dependent
Random Choice Lemma are used.National Science Foundatio
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