43 research outputs found
Research Problems from the BCC21
AbstractA collection of open problems, mostly presented at the problem session of the 21st British Combinatorial Conference
Permutation codes
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
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
For a function on the hypercube with Fourier expansion
, the hypercontractive inequality for
the noise operator allows bounding norms of in terms of norms of . If is Boolean-valued, the level-
inequality allows bounding the norm of in
terms of . 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 and to other settings where sharpness of the bound is
crucial. Also, no level- inequality for global functions over general
discrete product spaces is known.
We obtain sharp versions of the hypercontractive inequality and of the
level- 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 -- including hypercontractivity
and level- inequalities, character bounds, variants of Roth's theorem and of
Bogolyubov's lemma, and diameter bounds, that were obtained using our
techniques