On q-analogues and stability theorems

Abstract

Abstract. In this survey recent results about q-analogues of some classical theorems in extremal set theory are collected. They are related to determining the chromatic number of the q-analogues of Kneser graphs. For the proof one needs results on the number of 0-secant subspaces of point sets, so in the second part of the paper recent results on the structure of point sets having few 0-secant subspaces are discussed. Our attention is focussed on the planar case, where various stability results are given.Mathematics Subject Classification (2010). 05D05, 05B25.

On the stability of small blocking sets

“…When |B| is relatively far from q and gets closer to 3 2 (q + 1) then the bound on δ gets worse. Therefore, the aim of Sect.…”

“…3 is the following theorem. Theorem 1.5 Let B be a point set in PG(2, q), q ≥ 16, of size less than 3 2 (q + 1). Denote the number of 0-secants of B by δ, and assume that δ < min (q − 1) 2q + 1 − |B| 2(|B| − q) , 1 2 (q − √ q) 3/2 .…”

“…This result can easily be extended to sets of size q + k, (k ≤ √ q + 1), as follows. The proof can be found in [3].…”

“…(1) ( [4]) Assume that for the size of B, 3 2 q − 2 ≤ |B| ≤ 2q − 2 holds and δ < 2(2q − 1 − |B|). Then B can be obtained from a blocking set by deleting at most one point.…”

“…Many of them concentrate on small blocking sets of PG (2, q), these are blocking sets whose cardinality is less than 3 2 (q + 1). In some cases small minimal blocking sets are already characterized.…”