Michal Karonski (left) who built Poland’s probabilistic combinatorics group at Poznań, and a sculpture honoring the Polish mathematicians who first broke the Enigma machine (right, with David Conlon, picture taken by Jacob Fox).

I am visiting now Poznań for the 16th Conference on Random Structures and Algorithms. This bi-annually series of conferences started 30 years ago (as a satellite conference to the 1983 ICM which took place in Warsaw) and this time there was also a special celebration for Bela Bollobás 70th birthday. I was looking forward to this first visit to Poland which is, of course, a moving experience for me. Before Poznań I spent a few days in Gdańsk visiting Robert Alicki. Today (Wednesday) at the Poznań conference I gave a lecture on threshold phenomena and here are the slides. In the afternoon we had the traditional random run with a record number of runners. Let me briefly tell you about very few of the other lectures: **Update (Thursday):** A very good day, and among others a great talk of Jacob Fox on Relative Szemeredi Theorem (click for the slides from a similar talk from Budapest) where he presented a joint work with David Conlon and Yufei Zhao giving a very general and strong form of Szemeredi theorem for quasi-random sparse sets, which among other applications, leads to a much simpler proof of the Green -Tao theorem.

### Mathias Schacht

Mathias Schacht gave a wonderful talk on extremal results in random graphs (click for the slides) which describes some large recent body of highly successful research on the topic. Here are two crucial slides, and going through the whole presentation can give a very good overall picture.

### Vera Sós

Vera Sós gave an inspiring talk about the random nature of graphs which are extremal to the Ramsey property and connections with graph limits. Vera presented the following very interesting conjecture on graph limits. We say that a sequence of graphs has a** limit** if for every* k* and every graph *H* with *k* vertices the proportion in of induced *H*-subgraphs among all* k-*vertex induced subgraphs tend to a limit. Let us also say that has a ** V-limit** if for every

*k*and every

*e*the proportion in of induced subgraphs with k vertices and e edges among all

*k-*vertex induced subgraphs tend to a limit.

**Sós’ question**: Is having a V-limit equivalent to having a limit. This is open even in the case of quasirandomness, namely, when the limit is given by the Erdos-Renyi model

*G(n,p). (*Both a positive and a negative answer to this fundamental question would lead to many further (different) open problems.

**Update**: in this case V-limit is equivalent to limit, as several participants of the conference observed.)### Joel Spencer

Joel Spencer gave a great (blackboard) talk about algorithmic aspects of the probabilistic method, and how existence theorems via the probabilistic method now often require complicated randomized algorithm. Joel mentioned his famous six standard deviation theorem. In this case, Joel conjectured thirty years ago that there is no efficient algorithm to find the coloring promised by his theorem. Joel was delighted to see his conjecture being refuted first by Nikhil Bansal (who found an algorithm whose proof depends on the theorem) and then later by Shachar Lovett and Raghu Meka (who found a new algorithm giving a new proof) . In fact, Joel said, having his conjecture disproved is even more delightful than having it proved. Based on this experience Joel and I are now proposing another conjecture: **Kalai-Spencer (pre)conjecture:** Every existence statement proved by the probabilistic method can be complemented by an efficient (possibly randomized) algorithm. By “complemented by an efficient algorithm” we mean that there is an efficient(polynomial time) randomized algorithm to create the promised object with high probability. We refer to it as a preconjecture since the term “the probabilistic method” is not entirely well-defined. But it may be possible to put this conjecture on formal grounds, and to discuss it informally even before.

There is an old conjecture of Dick Karp that will certainly go against every formal version of the conjecture by Spencer and me. It is known that the clique number for a random graph in G(n,1/2) is 2 log n, and this has a simple and very basic probabilistic proof There is an easy polynomial time algorithm to find a clique of size log n and Karp conjectured in the mid 70s that finding a clique of size (1+t)log n in a random graph in G(n,1/2) is not in P!

I was wrong. The Karp conjecture is not going against my conjecture with Specer. We deal with finding efficiently objects guaranteed (in certainty) by probabilistic proofs, so we do not talk about decision problems like that in Karp’s conjecture, but about questions of the kind dealt with by Papadimitriou in his complexity classes.