71 research outputs found
A bound on the number of edges in graphs without an even cycle
We show that, for each fixed , an -vertex graph not containing a cycle
of length has at most edges.Comment: 16 pages, v2 appeared in Comb. Probab. Comp., v3 fixes an error in v2
and explains why the method in the paper cannot improve the power of k
further, v4 fixes the proof of Theorem 12 introduced in v
Negligible obstructions and Tur\'an exponents
We show that for every rational number of the form ,
where satisfy , there exists a graph such that the Tur\'an
number . Our result in particular
generates infinitely many new Tur\'an exponents. As a byproduct, we formulate a
framework that is taking shape in recent work on the Bukh--Conlon conjecture.Comment: 23 pages, 5 figures, v2 replaces Proposition 25 in v1, which contains
an error, this results in a weaker main theore
Ramsey-nice families of graphs
For a finite family of fixed graphs let be
the smallest integer for which every -coloring of the edges of the
complete graph yields a monochromatic copy of some . We
say that is -nice if for every graph with
and for every -coloring of there exists a
monochromatic copy of some . It is easy to see that if
contains no forest, then it is not -nice for any . It seems
plausible to conjecture that a (weak) converse holds, namely, for any finite
family of graphs that contains at least one forest, and for all
(or at least for infinitely many values of ),
is -nice. We prove several (modest) results in support of this
conjecture, showing, in particular, that it holds for each of the three
families consisting of two connected graphs with 3 edges each and observing
that it holds for any family containing a forest with at most 2
edges. We also study some related problems and disprove a conjecture by
Aharoni, Charbit and Howard regarding the size of matchings in regular
3-partite 3-uniform hypergraphs.Comment: 20 pages, 2 figure
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