9,925 research outputs found
Multiply intersecting families of sets
AbstractLet [n] denote the set {1,2,…,n}, 2[n] the collection of all subsets of [n] and F⊂2[n] be a family. The maximum of |F| is studied if any r subsets have an at least s-element intersection and there are no ℓ subsets containing t+1 common elements. We show that |F|⩽∑i=0t−sn−si+t+ℓ−st+2−sn−st+1−s+ℓ−2 and this bound is asymptotically the best possible as n→∞ and t⩾2s⩾2, r,ℓ⩾2 are fixed
Magnetic Monopoles and Free Fractionally Charged States at Accelerators and in Cosmic Rays
Unified theories of strong, weak and electromagnetic interactions which have
electric charge quantization predict the existence of topologically stable
magnetic monopoles. Intermediate scale monopoles are comparable with detection
energies of cosmic ray monopoles at IceCube and other cosmic ray experiments.
Magnetic monopoles in some models can be significantly lighter and carry two,
three or possibly even higher quanta of the Dirac magnetic charge. They could
be light enough for their effects to be detected at the LHC either directly or
indirectly. An example based on a D-brane inspired (trinification) model with the monopole carrying three quanta of Dirac
magnetic charge is presented. These theories also predict the existence of
color singlet states with fractional electric charge which may be accessible at
the LHC.Comment: 18 pages, 2 figures, minor revisions, references adde
Some New Bounds For Cover-Free Families Through Biclique Cover
An cover-free family is a family of subsets of a finite set
such that the intersection of any members of the family contains at least
elements that are not in the union of any other members. The minimum
number of elements for which there exists an with blocks is
denoted by .
In this paper, we show that the value of is equal to the
-biclique covering number of the bipartite graph whose vertices
are all - and -subsets of a -element set, where a -subset is
adjacent to an -subset if their intersection is empty. Next, we introduce
some new bounds for . For instance, we show that for
and
where is a constant satisfies the
well-known bound . Also, we
determine the exact value of for some values of . Finally, we
show that whenever there exists a Hadamard matrix of
order 4d
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