9,925 research outputs found

    Multiply intersecting families of sets

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    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

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    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 SU(3)C×SU(3)L×SU(3)RSU(3)_C\times SU(3)_L\times SU(3)_R (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

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    An (r,w;d)(r,w;d) cover-free family (CFF)(CFF) is a family of subsets of a finite set such that the intersection of any rr members of the family contains at least dd elements that are not in the union of any other ww members. The minimum number of elements for which there exists an (r,w;d)CFF(r,w;d)-CFF with tt blocks is denoted by N((r,w;d),t)N((r,w;d),t). In this paper, we show that the value of N((r,w;d),t)N((r,w;d),t) is equal to the dd-biclique covering number of the bipartite graph It(r,w)I_t(r,w) whose vertices are all ww- and rr-subsets of a tt-element set, where a ww-subset is adjacent to an rr-subset if their intersection is empty. Next, we introduce some new bounds for N((r,w;d),t)N((r,w;d),t). For instance, we show that for rwr\geq w and r2r\geq 2 N((r,w;1),t)c(r+ww+1)+(r+w1w+1)+3(r+w4w2)logrlog(tw+1), N((r,w;1),t) \geq c{{r+w\choose w+1}+{r+w-1 \choose w+1}+ 3 {r+w-4 \choose w-2} \over \log r} \log (t-w+1), where cc is a constant satisfies the well-known bound N((r,1;1),t)cr2logrlogtN((r,1;1),t)\geq c\frac{r^2}{\log r}\log t. Also, we determine the exact value of N((r,w;d),t)N((r,w;d),t) for some values of dd. Finally, we show that N((1,1;d),4d1)=4d1N((1,1;d),4d-1)=4d-1 whenever there exists a Hadamard matrix of order 4d
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