521 research outputs found

### Ramsey numbers and the size of graphs

For two graph H and G, the Ramsey number r(H, G) is the smallest positive
integer n such that every red-blue edge coloring of the complete graph K_n on n
vertices contains either a red copy of H or a blue copy of G. Motivated by
questions of Erdos and Harary, in this note we study how the Ramsey number
r(K_s, G) depends on the size of the graph G. For s \geq 3, we prove that for
every G with m edges, r(K_s,G) \geq c (m/\log m)^{\frac{s+1}{s+3}} for some
positive constant c depending only on s. This lower bound improves an earlier
result of Erdos, Faudree, Rousseau, and Schelp, and is tight up to a
polylogarithmic factor when s=3. We also study the maximum value of r(K_s,G) as
a function of m

### Small Complete Minors Above the Extremal Edge Density

A fundamental result of Mader from 1972 asserts that a graph of high average
degree contains a highly connected subgraph with roughly the same average
degree. We prove a lemma showing that one can strengthen Mader's result by
replacing the notion of high connectivity by the notion of vertex expansion.
Another well known result in graph theory states that for every integer t
there is a smallest real c(t) so that every n-vertex graph with c(t)n edges
contains a K_t-minor. Fiorini, Joret, Theis and Wood conjectured that if an
n-vertex graph G has (c(t)+\epsilon)n edges then G contains a K_t-minor of
order at most C(\epsilon)log n. We use our extension of Mader's theorem to
prove that such a graph G must contain a K_t-minor of order at most
C(\epsilon)log n loglog n. Known constructions of graphs with high girth show
that this result is tight up to the loglog n factor

### The number of additive triples in subsets of abelian groups

A set of elements of a finite abelian group is called sum-free if it contains
no Schur triple, i.e., no triple of elements $x,y,z$ with $x+y=z$. The study of
how large the largest sum-free subset of a given abelian group is had started
more than thirty years before it was finally resolved by Green and Ruzsa a
decade ago. We address the following more general question. Suppose that a set
$A$ of elements of an abelian group $G$ has cardinality $a$. How many Schur
triples must $A$ contain? Moreover, which sets of $a$ elements of $G$ have the
smallest number of Schur triples? In this paper, we answer these questions for
various groups $G$ and ranges of $a$.Comment: 20 pages; corrected the erroneous equality in (1) in the statement of
Theorem 1.

### Hamiltonicity, independence number, and pancyclicity

A graph on n vertices is called pancyclic if it contains a cycle of length l
for all 3 \le l \le n. In 1972, Erdos proved that if G is a Hamiltonian graph
on n > 4k^4 vertices with independence number k, then G is pancyclic. He then
suggested that n = \Omega(k^2) should already be enough to guarantee
pancyclicity. Improving on his and some other later results, we prove that
there exists a constant c such that n > ck^{7/3} suffices

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