347 research outputs found
The maximum number of copies in -free graphs
Generalizing Tur\'an's classical extremal problem, Alon and Shikhelman
investigated the problem of maximizing the number of copies in an -free
graph, for a pair of graphs and . Whereas Alon and Shikhelman were
primarily interested in determining the order of magnitude for large classes of
graphs , we focus on the case when and are paths, where we find
asymptotic and in some cases exact results. We also consider other structures
like stars and the set of cycles of length at least , where we derive
asymptotically sharp estimates. Our results generalize well-known extremal
theorems of Erd\H{o}s and Gallai
Ramsey numbers of Berge-hypergraphs and related structures
For a graph , a hypergraph is called a Berge-,
denoted by , if there exists a bijection such
that for every , . Let the Ramsey number
be the smallest integer such that for any -edge-coloring of
a complete -uniform hypergraph on vertices, there is a monochromatic
Berge- subhypergraph. In this paper, we show that the 2-color Ramsey number
of Berge cliques is linear. In particular, we show that for and where is a Berge-
hypergraph. For higher uniformity, we show that for
and for and sufficiently large. We
also investigate the Ramsey number of trace hypergraphs, suspension hypergraphs
and expansion hypergraphs.Comment: Updated to include suggestions of the refere
Inverse Tur\'an numbers
For given graphs and , the Tur\'an number is defined to be
the maximum number of edges in an -free subgraph of . Foucaud,
Krivelevich and Perarnau and later independently Briggs and Cox introduced a
dual version of this problem wherein for a given number , one maximizes the
number of edges in a host graph for which .
Addressing a problem of Briggs and Cox, we determine the asymptotic value of
the inverse Tur\'an number of the paths of length and and provide an
improved lower bound for all paths of even length. Moreover, we obtain bounds
on the inverse Tur\'an number of even cycles giving improved bounds on the
leading coefficient in the case of . Finally, we give multiple conjectures
concerning the asymptotic value of the inverse Tur\'an number of and
, suggesting that in the latter problem the asymptotic behavior
depends heavily on the parity of .Comment: updated to include the suggestions of reviewer
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