92 research outputs found
Large unavoidable subtournaments
Let denote the tournament on vertices consisting of three disjoint
vertex classes and of size , each of which is oriented as a
transitive subtournament, and with edges directed from to , from
to and from to . Fox and Sudakov proved that given a
natural number and there is such that
every tournament of order which is -far from
being transitive contains as a subtournament. Their proof showed that
and they conjectured that
this could be reduced to . Here we
prove this conjecture.Comment: 9 page
Forbidding intersection patterns between layers of the cube
A family is said to be an antichain
if for all distinct . A classic result
of Sperner shows that such families satisfy , which is easily seen to be best possible. One can
view the antichain condition as a restriction on the intersection sizes between
sets in different layers of . More generally one can ask,
given a collection of intersection restrictions between the layers, how large
can families respecting these restrictions be? Answering a question of Kalai,
we show that for most collections of such restrictions, layered families are
asymptotically largest. This extends results of Leader and the author.Comment: 16 page
Forbidden vector-valued intersections
We solve a generalised form of a conjecture of Kalai motivated by attempts to
improve the bounds for Borsuk's problem. The conjecture can be roughly
understood as asking for an analogue of the Frankl-R\"odl forbidden
intersection theorem in which set intersections are vector-valued. We discover
that the vector world is richer in surprising ways: in particular, Kalai's
conjecture is false, but we prove a corrected statement that is essentially
best possible, and applies to a considerably more general setting. Our methods
include the use of maximum entropy measures, VC-dimension, Dependent Random
Choice and a new correlation inequality for product measures.Comment: 40 page
Forbidding a Set Difference of Size 1
How large can a family \cal A \subset \cal P [n] be if it does not contain
A,B with |A\setminus B| = 1? Our aim in this paper is to show that any such
family has size at most \frac{2+o(1)}{n} \binom {n}{\lfloor n/2\rfloor }. This
is tight up to a multiplicative constant of . We also obtain similar results
for families \cal A \subset \cal P[n] with |A\setminus B| \neq k, showing that
they satisfy |{\mathcal A}| \leq \frac{C_k}{n^k}\binom {n}{\lfloor n/2\rfloor
}, where C_k is a constant depending only on k.Comment: 8 pages. Extended to include bound for families \cal A \subset \cal P
[n] satisfying |A\setminus B| \neq k for all A,B \in \cal
Random walks on quasirandom graphs
Let G be a quasirandom graph on n vertices, and let W be a random walk on G
of length alpha n^2. Must the set of edges traversed by W form a quasirandom
graph? This question was asked by B\"ottcher, Hladk\'y, Piguet and Taraz. Our
aim in this paper is to give a positive answer to this question. We also prove
a similar result for random embeddings of trees.Comment: 19 pages, 2 figure
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