20,880 research outputs found
On symmetric intersecting families
We make some progress on a question of Babai from the 1970s, namely: for with , what is the largest possible cardinality
of an intersecting family of -element subsets of
admitting a transitive group of automorphisms? We give upper and lower bounds
for , and show in particular that as if and only if for some function
that increases without bound, thereby determining the threshold
at which `symmetric' intersecting families are negligibly small compared to the
maximum-sized intersecting families. We also exhibit connections to some basic
questions in group theory and additive number theory, and pose a number of
problems.Comment: Minor change to the statement (and proof) of Theorem 1.4; the authors
thank Nathan Keller and Omri Marcus for pointing out a mistake in the
previous versio
Partially Symmetric Functions are Efficiently Isomorphism-Testable
Given a function f: {0,1}^n \to {0,1}, the f-isomorphism testing problem
requires a randomized algorithm to distinguish functions that are identical to
f up to relabeling of the input variables from functions that are far from
being so. An important open question in property testing is to determine for
which functions f we can test f-isomorphism with a constant number of queries.
Despite much recent attention to this question, essentially only two classes of
functions were known to be efficiently isomorphism testable: symmetric
functions and juntas.
We unify and extend these results by showing that all partially symmetric
functions---functions invariant to the reordering of all but a constant number
of their variables---are efficiently isomorphism-testable. This class of
functions, first introduced by Shannon, includes symmetric functions, juntas,
and many other functions as well. We conjecture that these functions are
essentially the only functions efficiently isomorphism-testable.
To prove our main result, we also show that partial symmetry is efficiently
testable. In turn, to prove this result we had to revisit the junta testing
problem. We provide a new proof of correctness of the nearly-optimal junta
tester. Our new proof replaces the Fourier machinery of the original proof with
a purely combinatorial argument that exploits the connection between sets of
variables with low influence and intersecting families.
Another important ingredient in our proofs is a new notion of symmetric
influence. We use this measure of influence to prove that partial symmetry is
efficiently testable and also to construct an efficient sample extractor for
partially symmetric functions. We then combine the sample extractor with the
testing-by-implicit-learning approach to complete the proof that partially
symmetric functions are efficiently isomorphism-testable.Comment: 22 page
A New Multilayered PCP and the Hardness of Hypergraph Vertex Cover
Given a -uniform hyper-graph, the E-Vertex-Cover problem is to find the
smallest subset of vertices that intersects every hyper-edge. We present a new
multilayered PCP construction that extends the Raz verifier. This enables us to
prove that E-Vertex-Cover is NP-hard to approximate within factor
for any and any . The result is
essentially tight as this problem can be easily approximated within factor .
Our construction makes use of the biased Long-Code and is analyzed using
combinatorial properties of -wise -intersecting families of subsets
ABJ(M) and Fractional M2's with Fractional M2 Charge
Recently Aharony, Bergman and Jafferis (ABJ) have argued that a 3d
U(N+M)xU(N) Chern-Simons gauge theory at level (k,-k) may have a vacuum with
N=6 supersymmetry only if M<k+1 and if a certain period of the B-field in a IIA
background is quantized. We use a braneology argument to argue that N=3
supersymmetry may be preserved under the weaker condition that 2Nk>M(M-k)-1
with no restriction on the B-field. IIB brane cartoons and 11d supergravity
solutions corresponding to N=3 vacua that do not preserve N=6 supersymmetry are
argued to represent cascading gauge theories, generalizing the N=2 Seiberg
duality conjectured by Giveon and Kutasov. While as usual the M2-brane charge
runs as a result of the twisted Bianchi identity for *G_4, the M5-brane charge
running relies on the fact that it wraps a torsion homology cycle.Comment: 16 pages, 3 eps figure
Systoles and kissing numbers of finite area hyperbolic surfaces
We study the number and the length of systoles on complete finite area
orientable hyperbolic surfaces. In particular, we prove upper bounds on the
number of systoles that a surface can have (the so-called kissing number for
hyperbolic surfaces). Our main result is a bound which only depends on the
topology of the surface and which grows subquadratically in the genus.Comment: A minor mistake and a computation fixed, small changes in the
exposition. 23 pages, 13 figure
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