395 research outputs found
Intersecting families of discrete structures are typically trivial
The study of intersecting structures is central to extremal combinatorics. A
family of permutations is \emph{-intersecting} if
any two permutations in agree on some indices, and is
\emph{trivial} if all permutations in agree on the same
indices. A -uniform hypergraph is \emph{-intersecting} if any two of its
edges have vertices in common, and \emph{trivial} if all its edges share
the same vertices.
The fundamental problem is to determine how large an intersecting family can
be. Ellis, Friedgut and Pilpel proved that for sufficiently large with
respect to , the largest -intersecting families in are the trivial
ones. The classic Erd\H{o}s--Ko--Rado theorem shows that the largest
-intersecting -uniform hypergraphs are also trivial when is large. We
determine the \emph{typical} structure of -intersecting families, extending
these results to show that almost all intersecting families are trivial. We
also obtain sparse analogues of these extremal results, showing that they hold
in random settings.
Our proofs use the Bollob\'as set-pairs inequality to bound the number of
maximal intersecting families, which can then be combined with known stability
theorems. We also obtain similar results for vector spaces.Comment: 19 pages. Update 1: better citation of the Gauy--H\`an--Oliveira
result. Update 2: corrected statement of the unpublished Hamm--Kahn result,
and slightly modified notation in Theorem 1.6 Update 3: new title, updated
citations, and some minor correction
Quantum logic is undecidable
We investigate the first-order theory of closed subspaces of complex Hilbert
spaces in the signature , where `' is the
orthogonality relation. Our main result is that already its quasi-identities
are undecidable: there is no algorithm to decide whether an implication between
equations and orthogonality relations implies another equation. This is a
corollary of a recent result of Slofstra in combinatorial group theory. It
follows upon reinterpreting that result in terms of the hypergraph approach to
quantum contextuality, for which it constitutes a proof of the inverse sandwich
conjecture. It can also be interpreted as stating that a certain quantum
satisfiability problem is undecidable.Comment: 11 pages. v3: improved exposition. v4: minor clarification
N=4 Multi-Particle Mechanics, WDVV Equation and Roots
We review the relation of N=4 superconformal multi-particle models on the
real line to the WDVV equation and an associated linear equation for two
prepotentials, F and U. The superspace treatment gives another variant of the
integrability problem, which we also reformulate as a search for closed flat
Yang-Mills connections. Three- and four-particle solutions are presented. The
covector ansatz turns the WDVV equation into an algebraic condition, for which
we give a formulation in terms of partial isometries. Three ideas for
classifying WDVV solutions are developed: ortho-polytopes, hypergraphs, and
matroids. Various examples and counterexamples are displayed
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