41 research outputs found
Global hypercontractivity and its applications
The hypercontractive inequality on the discrete cube plays a crucial role in
many fundamental results in the Analysis of Boolean functions, such as the KKL
theorem, Friedgut's junta theorem and the invariance principle. In these
results the cube is equipped with the uniform measure, but it is desirable,
particularly for applications to the theory of sharp thresholds, to also obtain
such results for general -biased measures. However, simple examples show
that when , there is no hypercontractive inequality that is strong
enough.
In this paper, we establish an effective hypercontractive inequality for
general that applies to `global functions', i.e. functions that are not
significantly affected by a restriction of a small set of coordinates. This
class of functions appears naturally, e.g. in Bourgain's sharp threshold
theorem, which states that such functions exhibit a sharp threshold. We
demonstrate the power of our tool by strengthening Bourgain's theorem, thereby
making progress on a conjecture of Kahn and Kalai and by establishing a
-biased analog of the invariance principle.
Our results have significant applications in Extremal Combinatorics. Here we
obtain new results on the Tur\'an number of any bounded degree uniform
hypergraph obtained as the expansion of a hypergraph of bounded uniformity.
These are asymptotically sharp over an essentially optimal regime for both the
uniformity and the number of edges and solve a number of open problems in the
area. In particular, we give general conditions under which the crosscut
parameter asymptotically determines the Tur\'an number, answering a question of
Mubayi and Verstra\"ete. We also apply the Junta Method to refine our
asymptotic results and obtain several exact results, including proofs of the
Huang--Loh--Sudakov conjecture on cross matchings and the
F\"uredi--Jiang--Seiver conjecture on path expansions.Comment: Subsumes arXiv:1906.0556
Covering and tiling hypergraphs with tight cycles
Given , we say that a -uniform hypergraph is a
tight cycle on vertices if there is a cyclic ordering of the vertices of
such that every consecutive vertices under this ordering form an
edge. We prove that if and , then every -uniform
hypergraph on vertices with minimum codegree at least has
the property that every vertex is covered by a copy of . Our result is
asymptotically best possible for infinitely many pairs of and , e.g.
when and are coprime.
A perfect -tiling is a spanning collection of vertex-disjoint copies
of . When is divisible by , the problem of determining the
minimum codegree that guarantees a perfect -tiling was solved by a
result of Mycroft. We prove that if and is not divisible
by and divides , then every -uniform hypergraph on vertices
with minimum codegree at least has a perfect
-tiling. Again our result is asymptotically best possible for infinitely
many pairs of and , e.g. when and are coprime with even.Comment: Revised version, accepted for publication in Combin. Probab. Compu
Sparse graphs with bounded induced cycle packing number have logarithmic treewidth
A graph is -free if it does not contain pairwise vertex-disjoint and
non-adjacent cycles. We show that Maximum Independent Set and 3-Coloring in
-free graphs can be solved in quasi-polynomial time. As a main technical
result, we establish that "sparse" (here, not containing large complete
bipartite graphs as subgraphs) -free graphs have treewidth (even, feedback
vertex set number) at most logarithmic in the number of vertices. This is
proven sharp as there is an infinite family of -free graphs without
-subgraph and whose treewidth is (at least) logarithmic.
Other consequences include that most of the central NP-complete problems
(such as Maximum Independent Set, Minimum Vertex Cover, Minimum Dominating Set,
Minimum Coloring) can be solved in polynomial time in sparse -free graphs,
and that deciding the -freeness of sparse graphs is polynomial time
solvable.Comment: 28 pages, 6 figures. v3: improved complexity result