365 research outputs found
Tur\'annical hypergraphs
This paper is motivated by the question of how global and dense restriction
sets in results from extremal combinatorics can be replaced by less global and
sparser ones. The result we consider here as an example is Turan's theorem,
which deals with graphs G=([n],E) such that no member of the restriction set
consisting of all r-tuples on [n] induces a copy of K_r.
Firstly, we examine what happens when this restriction set is replaced just
by all r-tuples touching a given m-element set. That is, we determine the
maximal number of edges in an n-vertex such that no K_r hits a given vertex
set.
Secondly, we consider sparse random restriction sets. An r-uniform hypergraph
R on vertex set [n] is called Turannical (respectively epsilon-Turannical), if
for any graph G on [n] with more edges than the Turan number ex(n,K_r)
(respectively (1+\eps)ex(n,K_r), no hyperedge of R induces a copy of K_r in G.
We determine the thresholds for random r-uniform hypergraphs to be Turannical
and to epsilon-Turannical.
Thirdly, we transfer this result to sparse random graphs, using techniques
recently developed by Schacht [Extremal results for random discrete structures]
to prove the Kohayakawa-Luczak-Rodl Conjecture on Turan's theorem in random
graphs.Comment: 33 pages, minor improvements thanks to two referee
Coloring random graphs online without creating monochromatic subgraphs
Consider the following random process: The vertices of a binomial random
graph are revealed one by one, and at each step only the edges
induced by the already revealed vertices are visible. Our goal is to assign to
each vertex one from a fixed number of available colors immediately and
irrevocably without creating a monochromatic copy of some fixed graph in
the process. Our first main result is that for any and , the threshold
function for this problem is given by , where
denotes the so-called \emph{online vertex-Ramsey density} of
and . This parameter is defined via a purely deterministic two-player game,
in which the random process is replaced by an adversary that is subject to
certain restrictions inherited from the random setting. Our second main result
states that for any and , the online vertex-Ramsey density
is a computable rational number. Our lower bound proof is algorithmic, i.e., we
obtain polynomial-time online algorithms that succeed in coloring as
desired with probability for any .Comment: some minor addition
Triangle-Intersecting Families of Graphs
A family of graphs F is said to be triangle-intersecting if for any two
graphs G,H in F, the intersection of G and H contains a triangle. A conjecture
of Simonovits and Sos from 1976 states that the largest triangle-intersecting
families of graphs on a fixed set of n vertices are those obtained by fixing a
specific triangle and taking all graphs containing it, resulting in a family of
size (1/8) 2^{n choose 2}. We prove this conjecture and some generalizations
(for example, we prove that the same is true of odd-cycle-intersecting
families, and we obtain best possible bounds on the size of the family under
different, not necessarily uniform, measures). We also obtain stability
results, showing that almost-largest triangle-intersecting families have
approximately the same structure.Comment: 43 page
A composition theorem for the Fourier Entropy-Influence conjecture
The Fourier Entropy-Influence (FEI) conjecture of Friedgut and Kalai [FK96]
seeks to relate two fundamental measures of Boolean function complexity: it
states that holds for every Boolean function , where
denotes the spectral entropy of , is its total influence,
and is a universal constant. Despite significant interest in the
conjecture it has only been shown to hold for a few classes of Boolean
functions.
Our main result is a composition theorem for the FEI conjecture. We show that
if are functions over disjoint sets of variables satisfying the
conjecture, and if the Fourier transform of taken with respect to the
product distribution with biases satisfies the conjecture,
then their composition satisfies the conjecture. As
an application we show that the FEI conjecture holds for read-once formulas
over arbitrary gates of bounded arity, extending a recent result [OWZ11] which
proved it for read-once decision trees. Our techniques also yield an explicit
function with the largest known ratio of between and
, improving on the previous lower bound of 4.615
Intersecting Families of Permutations
A set of permutations is said to be {\em k-intersecting} if
any two permutations in agree on at least points. We show that for any
, if is sufficiently large depending on , then the
largest -intersecting subsets of are cosets of stabilizers of
points, proving a conjecture of Deza and Frankl. We also prove a similar result
concerning -cross-intersecting subsets. Our proofs are based on eigenvalue
techniques and the representation theory of the symmetric group.Comment: 'Erratum' section added. Yuval Filmus has recently pointed out that
the 'Generalised Birkhoff theorem', Theorem 29, is false for k > 1, and so is
Theorem 27 for k > 1. An alternative proof of the equality part of the
Deza-Frankl conjecture is referenced, bypassing the need for Theorems 27 and
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