Jacques Tits invented generalized polygons to give a geometrical interpretation of the exceptional groups of Lie type. The prototype of these incidence geometries already appears in his 1956 paper, while they are axiomatically defined in his influential 1959 paper on triality. The finite versions of these objects have played an important role in the study of finite simple groups and in the general the theory of permutation groups. In the 70’s and 80’s, generalized polygons also started being studied under the theory of distance regular graphs (and more broadly association schemes). The theory of these geometries has been developed for its own sake as well (see this and this), and there are several long standing open problems about them that have attracted the attention of many mathematicians. In this post I want to discuss the deep connection that generalized polygons have had with extremal combinatorics, specifically with Ramsey theory and Turán type theorems.
Finite projective planes, which are in fact a special case of generalized polygons known as generalized triangles, have always been deeply connected to extremal combinatorics. They give us the optimal examples for several extremal problems on both graphs and hypergraphs. For example, they were behind the construction of Esther Klein, mentioned in the 1938 paper of Paul Erdős, which in fact determines the maximum number of edges in a -free graph on vertices asymptotically (it’s fun to look at the original number theoretic problem which motivated this construction and also the fact that there is no mention of the term ‘projective planes’ even though the concept is there). They are also present in the early constructions for some special cases of the classical Zarankiewicz problem. These early results form the foundation of extremal graph theory.
Finite generalized quadrangles (GQ) have played a similar role, but I have noticed that many times they don’t get mentioned explicitly by the extremal combinatorics community. There appears to be a gap in knowledge, which is probably because of the slightly more involved constructions of classical GQ’s. The so-called Benson graph of girth , that give us the asymptotic result for the Turán number of , is simply the incidence graph of a GQ. While Benson doesn’t say that explicitly in his 1966 paper, it’d be very surprising if he actually din’t know it as his PhD thesis was called Generalized Quadrangles and (B,N) Pairs. Similarly, the so-called Wenger graph from 1991, is just an induced subgraph of the incidence graph of a well known GQ, even though there is no mention of generalized quadrangles in Wenger’s paper. See my earlier blog post on Wenger graph for more details on this. Once you see that, it should be clear how Kopparty’s graph is related to the same GQ. Thus, the best known construction of optimal pseudorandom triangle-free graphs can be obtained via generalized quadrangles! This was also shown by Conlon, who started with the collinearity graph of a generalized quadrangle and gave a random construction from it that resulted in almost optimal triangle-free pseudorandom graphs. The basic idea behind these two constructions is in fact the same, just that Kopparty uses an algebraic method to achieve the end goal whereas Conlon uses a probabilistic one (which is why he ends up with an almost optimal graph).
These optimal triangle-free pseudorandom graphs give us the best known constructive lower bounds on the Ramsey number . As these can be obtained via generalized quadrangles, we see an application of GQ’s in Ramsey theory. Interestingly, GQ’s can also be used to give us the best known constructive lower bounds on the Ramsey number , as shown by Kostochka, Pudlák and Rödl. They have even been used in some hypergraph Ramsey problems to obtain tight lower bounds that beat the probabilistic ones (see Kostochka-Mubayi-Verstraëte). There are some other problems in Ramsey theory where GQ’s have shown up as well, see for example this and this.
In my recent work with John Bamberg and Thomas Lesgourgues on a minimal Ramsey problem we have also used generalized quadrangles. We noticed that a particular group theoretic model of GQ’s, due to Kantor, was perfectly suitable for this problem. I am certain that this idea will be useful in other problems in extremal combinatorics problems as well. I also believe that new interesting applications of generalized hexagons and octagons, as in the recent work of Mubayi and Verstraete (also described here), are just waiting to be found. One of the necessary steps for this is to learn more about these beautiful mathematical objects, and for that I would like to end with a few references.