237 research outputs found
An extension of Tur\'an's Theorem, uniqueness and stability
We determine the maximum number of edges of an -vertex graph with the
property that none of its -cliques intersects a fixed set .
For , the -partite Turan graph turns out to be the unique
extremal graph. For , there is a whole family of extremal graphs,
which we describe explicitly. In addition we provide corresponding stability
results.Comment: 12 pages, 1 figure; outline of the proof added and other referee's
comments incorporate
Edge-decompositions of graphs with high minimum degree
A fundamental theorem of Wilson states that, for every graph , every
sufficiently large -divisible clique has an -decomposition. Here a graph
is -divisible if divides and the greatest common divisor
of the degrees of divides the greatest common divisor of the degrees of
, and has an -decomposition if the edges of can be covered by
edge-disjoint copies of . We extend this result to graphs which are
allowed to be far from complete. In particular, together with a result of
Dross, our results imply that every sufficiently large -divisible graph of
minimum degree at least has a -decomposition. This
significantly improves previous results towards the long-standing conjecture of
Nash-Williams that every sufficiently large -divisible graph with minimum
degree at least has a -decomposition. We also obtain the
asymptotically correct minimum degree thresholds of for the
existence of a -decomposition, and of for the existence of a
-decomposition, where . Our main contribution is a
general `iterative absorption' method which turns an approximate or fractional
decomposition into an exact one. In particular, our results imply that in order
to prove an asymptotic version of Nash-Williams' conjecture, it suffices to
show that every -divisible graph with minimum degree at least
has an approximate -decomposition,Comment: 41 pages. This version includes some minor corrections, updates and
improvement
Semi-algebraic Ramsey numbers
Given a finite point set , a -ary semi-algebraic
relation on is the set of -tuples of points in , which is
determined by a finite number of polynomial equations and inequalities in
real variables. The description complexity of such a relation is at most if
the number of polynomials and their degrees are all bounded by . The Ramsey
number is the minimum such that any -element point set
in equipped with a -ary semi-algebraic relation , such
that has complexity at most , contains members such that every
-tuple induced by them is in , or members such that every -tuple
induced by them is not in .
We give a new upper bound for for and fixed.
In particular, we show that for fixed integers , establishing a subexponential upper bound on .
This improves the previous bound of due to Conlon, Fox, Pach,
Sudakov, and Suk, where is a very large constant depending on and
. As an application, we give new estimates for a recently studied
Ramsey-type problem on hyperplane arrangements in . We also study
multi-color Ramsey numbers for triangles in our semi-algebraic setting,
achieving some partial results
An extension of Turán's theorem, uniqueness and stability
We determine the maximum number of edges of an n -vertex graph G with the property that none of its r -cliques intersects a fixed set M⊂V(G) . For (r−1)|M|≥n , the (r−1) -partite Turán graph turns out to be the unique extremal graph. For (r−1)|M|<n , there is a whole family of extremal graphs, which we describe explicitly. In addition we provide corresponding stability results
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