44 research outputs found
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
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 RamseyâTurĂĄn 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
Lagrangians of Hypergraphs
The Lagrangian of a hypergraph is a function that in a sense seems to measure how âtightly packedâ a subset of the hypergraph one can find. Lagrangians were first used by Motzkin and Straus to obtain a new proof of a classic theorem of TurĂĄn, and subsequently found a number of very valuable applications in Extremal Hypergraph Theory; one remarkable result they yield is the disproof of a famous "jumping conjecture" of Erdos, which we reprove entirely; we will also introduce a very recent method based on Razborov's flag algebras to show that, though the jumping conjecture is false in general, hypergraphs "do jump" in some cases
Recent progress on bounds for sets with no three terms in arithmetic progression
This is the text accompanying my Bourbaki seminar on the work of Bloom and
Sisask, Croot, Lev, and Pach, and Ellenberg and Gijswijt.Comment: 33 pages; v2: several typos fixe
Explicit and quantitative results for abelian varieties over finite fields
Includes bibliographical references.2022 Fall.Let E be an ordinary elliptic curve over a prime field Fp. Attached to E is the characteristic polynomial of the Frobenius endomorphism, T2 â a1T + p, which controls several of the invariants of E, such as the point count and the size of the isogeny class. As we base change E over extensions Fpn, we may study the distribution of point counts for both of these invariants. Additionally, we look to quantify the rate at which these distributions converge to the expected distribution. More generally, one may consider these same questions for collections of ordinary elliptic curves and abelian varieties