First distribution invariants and EKR theorems
Let \(I_ n\) be the set of any of the following: (i) all points of a projective plane \(P_{n-1}\) of dimension n-1 over GF(q); (ii) all points of an affine space \(A_{n-1}\) of dimension n-1 over GF(q); (iii) elements of an n-set. A function \(a: I_ n\to R\) is called a weight function of \(I_ n\) if \(\sum_{v\in I_ n}a(v)\geq 0\). If W is a subspace (subset), let \(a(W)=\sum_{v\in W}a(v)\). Let \(\phi (a,d)=| \{W| a(W)\geq 0\}|\) where W's are (d-1)-dimensional subspaces or d-subsets. Consider the inequality \((*)\quad \phi (a,d)\geq N(I_ n,d)\) where \(N(I_ n,d)=\left[ \begin{matrix} n-1\\ d-1\end{matrix} \right]=(q^{n-1}- 1)...(q^{n-d+1}-1)/(q^ d-1)...(q-1)\) in case (i) and (ii), and \(N(I_ n,d)=\left( \begin{matrix} n-1\\ d-1\end{matrix} \right)\) in case (iii). The author proves (*) for (ii) and for (i) and (iii) under the condition \(d| n\). It is also proved that if \(d| n\), any set of \(\left[\begin{matrix} n- 1\\ d-1\end{matrix} \right]\) distinct (d-1)-subspaces in \(P_{n-1}\) contains a d-spread. These results are applied to study the first distribution invariant of \(J_ q(n,d)\), the q-analogue Johnson scheme.
scientific article; zbMATH DE number 4051645