### The critical window for the classical Ramsey-Turán problem

[Update: Po-Shen Loh gave a wonderful talk in Banff about our paper. Here is the video.]

Jacob Fox, Po-Shen Loh and I recently uploaded to the arXiv our new paper “The critical window for the classical Ramsey-Turán problem.” This paper revisits the following celebrated Ramsey-Turán result first proved by Szemerédi in 1972.

Theorem 1 (Szemerédi 1972) For every ${\epsilon > 0}$, there exists a ${\delta > 0}$ such that any graph on ${n}$ vertices containing no 4-clique and no independent set of size greater than ${\delta n}$ has at most ${(1/8 + \epsilon)n^2}$ edges.

Turán’s theorem tells us that if we just want to construct a ${K_4}$-free graph on ${n}$ vertices with the largest possible number of edges, then we should divide the ${n}$ vertices into three nearly-equal parts and put in all possible edges across parts. This yields roughly ${n^2/3}$ edges. However, this graph contains very large independent sets with ${n/3}$ vertices. So the independent set restriction (which gives the “Ramsey” part of Ramsey-Turán) rules out this construction and forces the optimal graphs to take on a very different structure.

At the time of Szemerédi’s 1972 result, most people believed that a ${K_4}$-free graph with largest independent set of size ${o(n)}$ has to be much more sparse, perhaps with only ${o(n^2)}$ edges. So it was a surprise when, four years later, Bollobás and Erdős gave a clever geometric construction, based on the isoperimetric inequality for the high dimensional sphere, that showed that the bound given by Szemerédi is essentially optimal.

Theorem 2 (Bollobás and Erdős 1976) There exists a ${K_4}$-free graph on ${n}$ vertices with largest independent set of size ${o(n)}$ and having ${(1/8 - o(1))n^2}$ edges.

Bollobás and Erdős asked what happens in the critical window, when the number of edges is about ${n^2/8}$. This problem has received considerable attention (e.g., it was featured in a paper by Erdős from 1990 titled “Some of my favourite unsolved problems”). One of the difficulties here is that Szemerédi’s original proof used the regularity lemma, which is a powerful tool but one that unfortunately gives very poor parameter dependencies. To obtain results for the Ramsey-Turán problem at the desired precision, a regularity-free proof is needed. We give a new proof of the classical Ramsey-Turán result that avoids the use of regularity. This allows us to obtain much better and in fact nearly-optimal dependencies for the classical Ramsey-Turán problem in the critical window near ${n^2/8}$ edges, and solve several longstanding open problems in this area.

### Extremal results for sparse pseudorandom graphs

David Conlon, Jacob Fox and I have just uploaded to the arXiv our paper Extremal results in sparse pseudorandom graphs. The main advance of this paper is a sparse extension of the counting lemma associated to Szemerédi’s regularity lemma, allowing us to extend a wide range of classical extremal and Ramsey results to sparse pseudorandom graphs.

An important trend in modern combinatorics research is in extending classical results to the sparse setting. For instance, Szemerédi’s theorem says that every subset of the integers with positive density contains arbitrarily long arithmetic progressions. The celebrated result of Green and Tao says that the primes also contain arbitrarily long arithmetic progressions. While the primes have zero density in the integers, they may be placed inside a pseudorandom set of “almost primes” with positive relative density. Green and Tao established a transference principle, allowing them to apply Szemerédi’s theorem as a black box to the sparse setting. Our work has a similar theme. We establish a transference principle extending many classical extremal graph theoretic results to sparse pseudorandom graphs.

One of the most powerful tools in extremal graph theory is Szemerédi’s regularity lemma. Roughly speaking, it says that every large graph can be partitioned into a bounded number of roughly equally-sized parts so that the graph is random-like between pairs of parts. With this tool in hand, many important results in extremal graph theory can be proven using a three-step recipe, known as the regularity method:

1. Starting with any graph ${G}$, apply Szemerédi’s regularity lemma to obtain a regular partition;
2. Clean up the graph and create an associated reduced graph. Solve an easier problem in the reduced graph;
3. Apply the counting lemma. Profit.

The counting lemma is a result that says that the number of embeddings of a fixed graph (e.g., a triangle) into the regular partition is roughly what you would expect if the large graph were actually random. The original version of Szemerédi’s regularity lemma is useful only for dense graphs. Kohayakawa and Rödl later independently developed regularity lemmas for sparse graphs. However, for sparse extensions of the applications, the counting lemma remained a key missing ingredient and an important open problem in the field. Our main advance lies in a counting lemma that complements the sparse regularity lemma.