The tensor power of the clique on @math vertices (denoted by @math ) is the graph on vertex set @math such that two vertices @math are connected if and only if @math for all @math . Let the density of a subset @math of @math to be @math , and let the vertex boundary of a set @math to be vertices which are incident to some vertex of @math , perhaps including points of @math . We investigate two similar problems on such graphs. First, we study the vertex isoperimetry problem. Given a density @math what is the smallest possible density of the vertex boundary of a subset of @math of density @math ? Let @math be the infimum of these minimum densities as @math . We find a recursive relation allows one to compute @math in time polynomial to the number of desired bits of precision. Second, we study given an independent set @math of density @math , how close it is to a maximum-sized independent set @math of density @math . We show that this deviation (measured by @math ) is at most @math as long as @math . This substantially improves on results of Alon, Dinur, Friedgut, and Sudakov (2004) and Ghandehari and Hatami (2008) which had an @math upper bound. We also show the exponent @math is optimal assuming @math tending to infinity and @math tending to @math . The methods have similarity to recent work by Ellis, Keller, and Lifshitz (2016) in the context of Kneser graphs and other settings. The author hopes that these results have potential applications in hardness of approximation, particularly in approximate graph coloring and independent set problems.
A result which also finds a tight'' super constant exponent @math for the independent set stability is proved in some very recent work @cite_24 @cite_10 @cite_4 @cite_6 @cite_18 @cite_16 on Kneser graphs and related structures. (See also @cite_14 and Proposition 4.3 of @cite_8 .) The techniques have high-level similarity to the ones adopted here: The author became aware of these similar proofs only after writing major portions of the manuscript. particularly in their use of compressions to prove a isoperimetric inequality which they then bootstrap to a combinatorial independent set stability result.
@cite_18 @cite_14 @cite_4 @cite_8 @cite_6 @cite_24 @cite_16 @cite_10
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We prove tight upper bounds on the logarithmic derivative of the independence and matching polynomials of d-regular graphs. For independent sets, this theorem is a strengthening of Kahn's result that a disjoint union of copies of Kd;d maximizes the number of independent sets of a bipartite d-regular graph, Galvin and Tet ali's result that the independence polynomial is maximized by the same, and Zhao's extension of both results to all d-regular graphs. For matchings, this shows that the matching polynomial and the total number of matchings of a d-regular graph are maximized by a union of copies of Kd;d. Using this we prove the asymptotic upper matching conjecture of Friedland, Krop, Lundow, and Markstrom. In probabilistic language, our main theorems state that for all d-regular graphs and all �, the occupancy fraction of the hard-core model and the edge occupancy fraction of the monomer-dimer model with fugacity � are maximized by Kd;d. Our method involves constrained optimization problems over distributions of random variables and applies to all d-regular graphs directly, without a reduction to the bipartite case. Using a variant of the method we prove a lower bound on the occupancy fraction of the hard-core model on any d-regular, vertex-transitive, bipartite graph: the occupancy fraction of such a graph is strictly greater than the occupancy fraction of the unique translationinvariant hard-core measure on the infinite d-regular tree A version of the Riesz-Sobolev convolution inequality is formulated and proved for arbitrary compact connected Abelian groups. Maximizers are characterized and a quantitative stability theorem is proved, under natural hypotheses. A corresponding stability theorem for sets whose sumset has nearly minimal measure is also proved, sharpening recent results of other authors. For the special case of the group @math , a continuous deformation of sets is developed, under which an appropriately scaled Riesz-Sobolev functional is shown to be nondecreasing. The full' edge isoperimetric inequality for the discrete cube (due to Harper, Bernstein, Lindsay and Hart) specifies the minimum size of the edge boundary @math of a set @math , as a function of @math . A weaker (but more widely-used) lower bound is @math , where equality holds iff @math is a subcube. In 2011, the first author obtained a sharp stability' version of the latter result, proving that if @math , then there exists a subcube @math such that @math . The weak' version of the edge isoperimetric inequality has the following well-known generalization for the @math -biased' measure @math on the discrete cube: if @math , or if @math and @math is monotone increasing, then @math . In this paper, we prove a sharp stability version of the latter result, which generalizes the aforementioned result of the first author. Namely, we prove that if @math , then there exists a subcube @math such that @math , where @math . This result is a central component in recent work of the authors proving sharp stability versions of a number of Erd o s-Ko-Rado type theorems in extremal combinatorics, including the seminal complete intersection theorem' of Ahlswede and Khachatrian. In addition, we prove a biased-measure analogue of the full' edge isoperimetric inequality, for monotone increasing sets, and we observe that such an analogue does not hold for arbitrary sets, hence answering a question of Kalai. We use this result to give a new proof of the full' edge isoperimetric inequality, one relying on the Kruskal-Katona theorem. We show that any language in nondeterministic time @math , where the number of iterated exponentials is an arbitrary function @math , can be decided by a multiprover interactive proof system with a classical polynomial-time verifier and a constant number of quantum entangled provers, with completeness @math and soundness @math , where the number of iterated exponentials is @math and @math is a universal constant. The result was previously known for @math and @math ; we obtain it for any time-constructible function @math . The result is based on a compression technique for interactive proof systems with entangled provers that significantly simplifies and strengthens a protocol compression result of Ji (STOC'17). As a separate consequence of this technique we obtain a different proof of Slofstra's recent result (unpublished) on the uncomputability of the entangled value of multiprover games. Finally, we show that even minor improvements to our compression result would yield remarkable consequences in computational complexity theory and the foundations of quantum mechanics: first, it would imply that the class MIP* contains all computable languages; second, it would provide a negative resolution to a multipartite version of Tsirelson's problem on the relation between the commuting operator and tensor product models for quantum correlations.