Ramsey Properties of Random Graphs; A Sharp Threshold Proven via A Hypergraph Regularity Lemma. Ehud Friedgut, Vojtech Rödl, Andrzej Rucinski, Prasad Tetali

Special thanks to the Tetali family for costume design.

Chapter I

We will say a graph is a Ramsey graph if every bi-coloring of its edges contains a monochromatic triangle. e.g. Why ?!

Is there a sharp threshold?

Theorem: Yes, there does.

Why is the critical edge probability? The expected number of triangles per edge is

Chapter II

A multi-partite graph on vertex sets is -regular if all but of the pairs are -regular Easy if k is very small or very large…

Szemerédi’s Regularity Lemma: Weighted variations? Sparse graphs? Hypergraphs?

A hitting set of a graph G is a set of vertices that intersects every edge. In a dense graph on vertices there may be hitting sets. We would like to capture all hitting sets by a family of cores so that: 1.Every hitting set contains a core. 2. The number of cores is. 3. Every core is of size linear in.

If G is a complete bipartite graph on vertex sets U, V take the cores to be U and V. If G is -regular bipartite take all sets U’ or V’ such that or UV 1.Every hitting set contains a core. 3. Every core is of size linear in. 2. The number of cores is.

In a general graph – fix a Szemerédi partition. Draw the super-graph of regular pairs. A core will be any set obtained by taking a hitting set in the super-graph and taking at least of the vertices in all the super-vertices involved.

Chapter III

“Theorem”: Sharp threshold Global property Coarse threshold Local property e.g. connectivity has a sharp threshold - whereas containing a triangle has a coarse threshold.

…which means exactly that Ramsiness has a sharp threshold! Any such would be sensitive to small global enhancement … If Ramsiness had a coarse threshold it would be local – a typical non-Ramsey would be sensitive to local perturbations…

Let be typical in. Assume is non-Ramsey. Assume there exists a small magical graph, say, such that Show that this implies

is not seen in ! What about ?

Many copies of will pose restrictions if they appear – e.g. a problematic copy: We can color But in every proper coloring of one of the following will happen:

Using probabilistic techniques we can arrange a large subset of these restrictions as follows: Every restriction consists of five elements such that every proper coloring must agree with on at least one of them. B B B R R

For every proper coloring, the set of (graph)edges of on which it agrees with is a hitting set of. This defines a hypergraph with (hypergraph)edges of size 5. : B B B R R Given a proper coloring of, and an edge of then there exists an element in for which agrees with.

How does one show sensitivity to global enhancement? Every large partial coloring survives the addition of a random copy of with probability. There are approximately colorings. Union bound : Depends on the value of ! There may be too many colorings.

Last chapter:

We have a hypergraph of restrictions such that every proper coloring defines a hitting set of. But, there are too many colorings. We would like to capture them by a family of cores such that : 2. The number of cores is. 1. Every hitting set contains a core. 3. Every core is of size. We then can improve the union bound by clumping:

There are many colorings : Survival probability of each. Colorings (hitting sets) Cores All these colorings share a core.

A Frankl – Rödl partition 2. Partition every one of the bipartite graphs formed into (non-induced) subgraphs. 1.Partition the vertices of (auxiliary partition)

Choosing five of these bipartite graphs and a subgraph of each gives a polyad, a set of 5 subsets (anologous to a pair of sets in a Szemerédi partition.)

The density of a polyad = The number of copies of belonging to (The total number of copies of ) A regular Polyad – every sufficiently “large” subgraph has density close to that of the polyad.

“Theorem”: If is a typical graph in and is the corresponding restriction hypergraph then there exists a Frankl-Rödl partition of such that “most” of the polyads formed are -regular. This enables us to define cores, capture all colorings efficiently and finish the proof.

So, what is the definition of a core? Believe me, you don’t want to know.

And In conclusion I would like to say:

Ramsiness has a sharp threshold because it is a global property. Union bounds can be improved by clumping Clumping can be done if the underlying structure has an inherent regularity. Frankl –Rödl type partitions can extract regularity from various hypergraphs. Thank you for your attention!!