58 research outputs found
Many -copies in graphs with a forbidden tree
For graphs and , let be the maximum
possible number of copies of in an -free graph on vertices. The
study of this function, which generalises the well-studied Tur\'an numbers of
graphs, was initiated recently by Alon and Shikhelman. We show that if is a
tree then for some integer , thus answering one of their questions.Comment: 9 pages, 1 figur
Radon Numbers for Trees
Many interesting problems are obtained by attempting to generalize classical
results on convexity in Euclidean spaces to other convexity spaces, in
particular to convexity spaces on graphs. In this paper we consider
-convexity on graphs. A set of vertices in a graph is -convex
if every vertex not in has at most one neighbour in . More specifically,
we consider Radon numbers for -convexity in trees.
Tverberg's theorem states that every set of points in
can be partitioned into sets with intersecting convex hulls.
As a special case of Eckhoff's conjecture, we show that a similar result holds
for -convexity in trees.
A set of vertices in a graph is called free, if no vertex of has
more than one neighbour in . We prove an inequality relating the Radon
number for -convexity in trees with the size of a maximal free set.Comment: 17 pages, 13 figure
Hypergraphs with no tight cycles
We show that every r-uniform hypergraph on n vertices which does not contain a tight cycle has has at most O(n^{r-1}(log n)^{5}) edges. This is an improvement on the previously best-known bound, of n^{r-1}e^{O(\sqrt{log n})} due to Sudakov and Tomon, and our proof builds on their work. A recent
construction of B. Janzer implies that our bound is tight up to an O((log n)^{4} log log n) factor
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