109 research outputs found
Quantum communication complexity of symmetric predicates
We completely (that is, up to a logarithmic factor) characterize the
bounded-error quantum communication complexity of every predicate
depending only on (). Namely, for a predicate
on let \ell_0(D)\df \max\{\ell : 1\leq\ell\leq n/2\land
D(\ell)\not\equiv D(\ell-1)\} and \ell_1(D)\df \max\{n-\ell : n/2\leq\ell <
n\land D(\ell)\not\equiv D(\ell+1)\}. Then the bounded-error quantum
communication complexity of is equal (again, up to a
logarithmic factor) to . In particular, the
complexity of the set disjointness predicate is . This result
holds both in the model with prior entanglement and without it.Comment: 20 page
On the Fon-der-Flaass Interpretation of Extremal Examples for Turan's (3,4)-problem
In 1941, Turan conjectured that the edge density of any 3-graph without
independent sets on 4 vertices (Turan (3,4)-graph) is >= 4/9(1-o(1)), and he
gave the first example witnessing this bound. Brown (1983) and Kostochka (1982)
found many other examples of this density. Fon-der-Flaass (1988) presented a
general construction that converts an arbitrary -free orgraph
into a Turan (3,4)-graph. He observed that all Turan-Brown-Kostochka
examples result from his construction, and proved the bound >= 3/7(1-o(1)) on
the edge density of any Turan (3,4)-graph obtainable in this way.
In this paper we establish the optimal bound 4/9(1-o(1)) on the edge density
of any Turan (3,4)-graph resulting from the Fon-der-Flaass construction under
any of the following assumptions on the undirected graph underlying the
orgraph :
1. is complete multipartite;
2. The edge density of is >= (2/3-epsilon) for some absolute constant
epsilon>0.
We are also able to improve Fon-der-Flaass's bound to 7/16(1-o(1)) without
any extra assumptions on
Diameter of Polyhedra: Limits of Abstraction
We investigate the diameter of a natural abstraction of the
-skeleton of polyhedra. Even if this abstraction is more general than
other abstractions previously studied in the literature,
known upper bounds on the diameter of polyhedra continue to hold
here. On the other hand, we show that this abstraction has its
limits by providing an almost quadratic lower bound
On the Number of Pentagons in Triangle-Free Graphs
Using the formalism of flag algebras, we prove that every triangle-free graph
with vertices contains at most cycles of length five.
Moreover, the equality is attained only when is divisible by five and
is the balanced blow-up of the pentagon. We also compute the maximal number of
pentagons and characterize extremal graphs in the non-divisible case provided
is sufficiently large. This settles a conjecture made by Erd\H{o}s in 1984.Comment: 16 pages, accepted to Journal of Combinatorial Theory Ser.
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