Intersection theorems and mod p rank of inclusion matrices
نویسنده
چکیده
Higher incidence matrices have proved an important tool both in design theory and extremal set theory. In the present paper some tight bounds on the rank over finite fields of some inclusion matrices are derived. In particular, a short proof of Wilson's mod p rank formula is given. A problem of Graham, Li, and Li concerning bases for so-called null t-designs is solved as well.
منابع مشابه
Some lower bounds for the $L$-intersection number of graphs
For a set of non-negative integers~$L$, the $L$-intersection number of a graph is the smallest number~$l$ for which there is an assignment of subsets $A_v subseteq {1,dots, l}$ to vertices $v$, such that every two vertices $u,v$ are adjacent if and only if $|A_u cap A_v|in L$. The bipartite $L$-intersection number is defined similarly when the conditions are considered only for the ver...
متن کاملSome rank equalities for finitely many tripotent matrices
A rank equality is established for the sum of finitely many tripotent matrices via elementary block matrix operations. Moreover, by using this equality and Theorems 8 and 10 in [Chen M. and et al. On the open problem related to rank equalities for the sum of finitely many idempotent matrices and its applications, The Scientific World Journal 2014 (2014), Article ID 702413, 7 page...
متن کاملA Remark on the Rank of Positive Semidefinite Matrices Subject to Affine Constraints
Let K n be the cone of positive semideenite n n matrices and let A be an aane subspace of the space of symmetric matrices such that the intersection K n \ A is non-empty and bounded. Suppose that n 3 and that codim A = ? r+2 2 for some 1 r n?2. Then there is a matrix X 2 K n \A such that rank X r. We give a short geometric proof of this result, use it to improve a bound on realizability of weig...
متن کاملStill Another Rank Determination of Set Intersection Matrices with an Application in Communication Complexity
The set-intersection function gives the cardinality of the intersection of two sets. In order to obtain a lower bound for the communication complexity of this function, the rank of the corresponding characteristic function-value matrices was calculated in 4] and 5]. In this note, the rank of these matrices is determined by another method of proof, which makes use of a factorization into a produ...
متن کاملOn higher rank numerical hulls of normal matrices
In this paper, some algebraic and geometrical properties of the rank$-k$ numerical hulls of normal matrices are investigated. A characterization of normal matrices whose rank$-1$ numerical hulls are equal to their numerical range is given. Moreover, using the extreme points of the numerical range, the higher rank numerical hulls of matrices of the form $A_1 oplus i A_2$, where $A_1...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- J. Comb. Theory, Ser. A
دوره 54 شماره
صفحات -
تاریخ انتشار 1990