43 research outputs found

    Research Problems from the BCC21

    Get PDF
    AbstractA collection of open problems, mostly presented at the problem session of the 21st British Combinatorial Conference

    Permutation codes

    Get PDF
    AbstractThere are many analogies between subsets and permutations of a set, and in particular between sets of subsets and sets of permutations. The theories share many features, but there are also big differences. This paper is a survey of old and new results about sets (and groups) of permutations, concentrating on the analogies and on the relations to coding theory. Several open problems are described

    A Log-Sobolev Inequality for the Multislice, with Applications

    Get PDF
    Let kappa in N_+^l satisfy kappa_1 + *s + kappa_l = n, and let U_kappa denote the multislice of all strings u in [l]^n having exactly kappa_i coordinates equal to i, for all i in [l]. Consider the Markov chain on U_kappa where a step is a random transposition of two coordinates of u. We show that the log-Sobolev constant rho_kappa for the chain satisfies rho_kappa^{-1} <= n * sum_{i=1}^l 1/2 log_2(4n/kappa_i), which is sharp up to constants whenever l is constant. From this, we derive some consequences for small-set expansion and isoperimetry in the multislice, including a KKL Theorem, a Kruskal - Katona Theorem for the multislice, a Friedgut Junta Theorem, and a Nisan - Szegedy Theorem

    Sharp Hypercontractivity for Global Functions

    Full text link
    For a function ff on the hypercube {0,1}n\{0,1\}^n with Fourier expansion f=βˆ‘SβŠ†[n]f^(S)Ο‡Sf=\sum_{S\subseteq[n]}\hat f(S)\chi_S, the hypercontractive inequality for the noise operator allows bounding norms of Tρf=βˆ‘Sρ∣S∣f^(S)Ο‡ST_\rho f=\sum_S\rho^{|S|}\hat f(S)\chi_S in terms of norms of ff. If ff is Boolean-valued, the level-dd inequality allows bounding the norm of f=d=βˆ‘βˆ£S∣=df^(S)Ο‡Sf^{=d}=\sum_{|S|=d}\hat f(S)\chi_S in terms of E[f]E[f]. These two inequalities play a central role in analysis of Boolean functions and its applications. While both inequalities hold in a sharp form when the hypercube is endowed with the uniform measure, they do not hold for more general discrete product spaces. Finding a `natural' generalization was a long-standing open problem. In [P. Keevash et al., Global hypercontractivity and its applications, J. Amer. Math. Soc., to appear], a hypercontractive inequality for this setting was presented, that holds for functions which are `global' -- namely, are not significantly affected by a restriction of a small set of coordinates. This hypercontractive inequality is not sharp, which precludes applications to the symmetric group SnS_n and to other settings where sharpness of the bound is crucial. Also, no level-dd inequality for global functions over general discrete product spaces is known. We obtain sharp versions of the hypercontractive inequality and of the level-dd inequality for this setting. Our inequalities open the way for diverse applications to extremal set theory and to group theory. We demonstrate this by proving quantitative bounds on the size of intersecting families of sets and vectors under weak symmetry conditions and by describing numerous applications to the study of functions on SnS_n -- including hypercontractivity and level-dd inequalities, character bounds, variants of Roth's theorem and of Bogolyubov's lemma, and diameter bounds, that were obtained using our techniques
    corecore