163 research outputs found
The Constructor-Blocker Game
We study the following game version of generalized graph Tur\'an problems.
For two fixed graphs and , two players, Constructor and Blocker,
alternately claim unclaimed edges of the complete graph . Constructor can
only claim edges so that he never claims all edges of any copy of , i.e.his
graph must remain -free, while Blocker can claim unclaimed edges without
restrictions. The game ends when Constructor cannot claim further edges or when
all edges have been claimed. The score of the game is the number of copies of
with all edges claimed by Constructor. Constructor's aim is to maximize the
score, while Blocker tries to keep the score as low as possible. We denote by
the score of the game when both players play optimally and
Constructor starts the game.
In this paper, we obtain the exact value of when both and
are stars and when , . We determine the asymptotics of
when is a star and is a tree and when , , and we derive
upper and lower bounds on
On the maximum number of odd cycles in graphs without smaller odd cycles
We prove that for each odd integer , every graph on vertices
without odd cycles of length less than contains at most cycles of
length . This generalizes the previous results on the maximum number of
pentagons in triangle-free graphs, conjectured by Erd\H{o}s in 1984, and
asymptotically determines the generalized Tur\'an number
for odd . In contrary to the previous results
on the pentagon case, our proof is not computer-assisted
High dimensional Hoffman bound and applications in extremal combinatorics
One powerful method for upper-bounding the largest independent set in a graph
is the Hoffman bound, which gives an upper bound on the largest independent set
of a graph in terms of its eigenvalues. It is easily seen that the Hoffman
bound is sharp on the tensor power of a graph whenever it is sharp for the
original graph.
In this paper, we introduce the related problem of upper-bounding independent
sets in tensor powers of hypergraphs. We show that many of the prominent open
problems in extremal combinatorics, such as the Tur\'an problem for
(hyper-)graphs, can be encoded as special cases of this problem. We also give a
new generalization of the Hoffman bound for hypergraphs which is sharp for the
tensor power of a hypergraph whenever it is sharp for the original hypergraph.
As an application of our Hoffman bound, we make progress on the problem of
Frankl on families of sets without extended triangles from 1990. We show that
if then the extremal family is the star,
i.e. the family of all sets that contains a given element. This covers the
entire range in which the star is extremal. As another application, we provide
spectral proofs for Mantel's theorem on triangle-free graphs and for
Frankl-Tokushige theorem on -wise intersecting families
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