The Intersection Problem for Kite-GDDs of Type $2^{u}$
Received:November 10, 2020  Revised:May 20, 2021
Key Words: kite-GDD   group divisible design   intersection number
Fund Project:Supported by the National Natural Science Foundation of China (Grant No.11601137) and the Research Program of Science and Technology at Universities of Inner Mongolia Autonomous Region (Grant Nos.NJZY19231; NJZZ21052).
 Author Name Affiliation Yonghong AN College of Continuing Education, Hulunbuir University, Inner Mongolia 021008, P. R. China Guizhi ZHANG Division of Science and Technology, Hulunbuir University, Inner Mongolia 021008, P. R. China
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The intersection problem for kite-GDDs is the determination of all pairs $(T,s)$ such that there exists a pair of kite-GDDs $(X,{\cal H},{\cal B}_1)$ and $(X,{\cal H},{\cal B}_2)$ of the same type $T$ satisfying $|{\cal B}_1\cap {\cal B}_2|=s$. In this paper the intersection problem for a pair of kite-GDDs of type $2^u$ is investigated. Let $J(u)=\{s:$ $\exists$ a pair of kite-GDDs of type $2^u$ intersecting in $s$ blocks$\}$; $I(u)=\{0,1,\ldots,b_{u}-2,b_{u}\}$, where $b_u=u(u-1)/2$ is the number of blocks of a kite-GDD of type $2^u$. We show that for any positive integer $u\geq 4$, $J(u)=I(u)$ and $J(3)= \{0,3\}$.