169 research outputs found
On positive hypergraphs
Camarena, Cs\'{o}ka, Hubai, Lippner, and Lov\'{a}sz introduced the notion of
positive graphs. This notion naturally extends to -uniform hypergraphs. In
the case when is odd, we prove that a hypergraph is positive if and only if
its Levi graph is positive. As an application, we show that the -subdivision
of is not a positive graph when is odd.Comment: final version accepted by European Journal of Combinatoric
On the asymptotic of lottery numbers
Let denote the minimum number of -subsets of an -set such
that all the -subsets are intersected by one of them in at
least elements. The case corresponds to the covering numbers, while
the case corresponds to the Tur\'an numbers. In both cases, there exists
a limit of as . We prove the existence
of this limit in the general case
Modified Erd\H{o}s-Ginzburg-Ziv constants for
Let be a finite abelian group written additively, and let be a
multiple of its exponent. The modified Erd\H{o}s-Ginzburg-Ziv constant
is the smallest integer such that every zero-sum
sequence of length over has a zero-sum subsequence of length . We
find exact values of for .Comment: 3 page
Non-three-colorable common graphs exist
A graph H is called common if the total number of copies of H in every graph
and its complement asymptotically minimizes for random graphs. A former
conjecture of Burr and Rosta, extending a conjecture of Erdos asserted that
every graph is common. Thomason disproved both conjectures by showing that the
complete graph of order four is not common. It is now known that in fact the
common graphs are very rare. Answering a question of Sidorenko and of Jagger,
Stovicek and Thomason from 1996 we show that the 5-wheel is common. This
provides the first example of a common graph that is not three-colorable.Comment: 9 page
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