Maximal cocliques in the Kneser graph on plane-solid flags in PG(6,q)

Abstract

For q ≥ 27 we determine the independence number α( ) of the Kneser graph on plane-solid flags in PG(6, q). More precisely we describe all maximal independent sets of size at least q 11 and show that every other maximal example has cardinality at most a constant times q 10 .

On the Chromatic Number of some generalized Kneser Graphs

“…Now, for the case covered in this work, that is in the Kneser graph qΓ 7,{3,4} , the required structural information as well as the bound α ′ is provided in [9]. Furthermore, we remark that the proof provided here does not only cover that case, but also every graph qΓ 2d+1,{d,d+1} , for which an analogous result to that given for d = 3 in [9] holds. We state this requirement more precisely in the following conjecture.…”

“…One reason for the lack of this structural information is, that in some cases the independence number was determined by algebraic arguments and these do not automatically give the structure of the largest independent sets, see for example the recent work [4]. Now, for the case covered in this work, that is in the Kneser graph qΓ 7,{3,4} , the required structural information as well as the bound α ′ is provided in [9]. Furthermore, we remark that the proof provided here does not only cover that case, but also every graph qΓ 2d+1,{d,d+1} , for which an analogous result to that given for d = 3 in [9] holds.…”

On the chromatic number of two generalized Kneser graphs

“…3. In [9] the authors classified the largest independent sets of the Kneser graph qK 7;{3,4} for q ≥ 27 and also proved an upper bound for the second largest maximal independent sets.…”

An algebraic approach to Erdős-Ko-Rado sets of flags in spherical buildings

“…• When n = 6, it was shown in [MW19] that a maximal set of EKR-flags of type {3, 4} has size of order q 11 . Our bound again falls an order of q 1/2 short and gives an upper bound of order q 23/2 .…”

“…And yet, researchers have investigated these kinds of problems as well, but with a different toolbox in hand. Notable examples in finite geometry are the papers due to Blokhuis and Brouwer [BB17], with Szőnyi [BBS14], with Güven [BBG14] and the paper by the last author and Werner [MW19]. All of these deal with flags in finite geometries, which are sets of pairwise incident elements of the geometry.…”