On generalizing generalized polygons

Abstract

The purpose of this paper is to reveal in geometric terms a decade-old construction of certain families of graphs with nice extremal properties. Construction of the graphs in question is motivated by the way in which regular generalized polygons may be embedded in their Lie algebras, so that point-line incidence corresponds to the vanishing Lie product. The only caveat is that the generalized polygons are greatly limited in number. By performing successive truncations on an infinite root system of type e A 1 , we are able to obtain an infinite series of incidence structures which approximate the behavior of generalized polygons. Indeed, the first two members of the series are exactly the affine parts of the generalized polygons of type A 2 and B 2 .

Turán Numbers of Hypergraph Suspensions of Even Cycles

“…Since probabilistic lower bounds for 3-graphs tend to be weak, we try to strengthen this result via a look at the graphs D(k, q). Here we present a summary of the properties of D(k, q); for more details, the reader is referred to [9,10,22].…”

“…For k ∈ {2, 3, 5}, the best known lower bounds are given by the bipartite graphs CD(k, q) [9,10] for integers k ≥ 2 and prime powers q. These graphs arise from Lie algebraic incidence structures that approximate the behavior of generalized polygons, and are analyzed in detail in [22]. For a recent survey on the even cycle problem, the reader is referred to [20].…”

Graphs from Generalized Kac--Moody Algebras

“…This is motivated by work in [6] and [17]. The basic construction is identical to the presentation in [17] with the exception of the Lie algebra used.…”

“…This is motivated by work in [6] and [17]. The basic construction is identical to the presentation in [17] with the exception of the Lie algebra used. In particular, the Cartan subalgebra differs in certain cases, thereby allowing us to obtain new families of graphs that would otherwise be degenerate.…”

“…The authors would like to thank Andrew Woldar for sharing a preprint of his paper [17] (now published). After careful scrutiny and conversations with Woldar, the claim in that paper that D(k, q) is given precisely by the Lie algebraic construction was found to be false for larger dimensions.…”