16 research outputs found
Complete intersection theorem and complete nontrivial-intersection theorem for systems of set partitions
We prove the complete intersection theorem and complete
nontrivial-intersection theorem for systems of set partitionsComment: add aknowlegments and distribute material between two my papers in
arxiv in another orde
Short proofs of three results about intersecting systems
In this note, we give short proofs of three theorems about intersection
problems. The first one is a determination of the maximum size of a nontrivial
-uniform, -wise intersecting family for , which improves upon a recent result of
O'Neill and Verstra\"{e}te. Our proof also extends to -wise,
-intersecting families, and from this result we obtain a version of the
Erd\H{o}s-Ko-Rado theorem for -wise, -intersecting families.
The second result partially proves a conjecture of Frankl and Tokushige about
-uniform families with restricted pairwise intersection sizes.
The third result concerns graph intersections. Answering a question of Ellis,
we construct -intersecting families of graphs which have size larger
than the Erd\H{o}s-Ko-Rado-type construction whenever is sufficiently large
in terms of .Comment: 12 pages; we added a new result, Theorem 1
Two-dimensional patterns with distinct differences; constructions, bounds, and maximal anticodes
A two-dimensional (2-D) grid with dots is called a configuration with distinct differences if any two lines which connect two dots are distinct either in their length or in their slope. These configurations are known to have many applications such as radar, sonar, physical alignment, and time-position synchronization. Rather than restricting dots to lie in a square or rectangle, as previously studied, we restrict the maximum distance between dots of the configuration; the motivation for this is a new application of such configurations to key distribution in wireless sensor networks. We consider configurations in the hexagonal grid as well as in the traditional square grid, with distances measured both in the Euclidean metric, and in the Manhattan or hexagonal metrics. We note that these configurations are confined inside maximal anticodes in the corresponding grid. We classify maximal anticodes for each diameter in each grid. We present upper bounds on the number of dots in a pattern with distinct differences contained in these maximal anticodes. Our bounds settle (in the negative) a question of Golomb and Taylor on the existence of honeycomb arrays of arbitrarily large size. We present constructions and lower bounds on the number of dots in configurations with distinct differences contained in various 2-D shapes (such as anticodes) by considering periodic configurations with distinct differences in the square grid