232 research outputs found
Excluding pairs of tournaments
The Erd\H{o}s-Hajnal conjecture states that for every given undirected graph
there exists a constant such that every graph that does not
contain as an induced subgraph contains a clique or a stable set of size at
least . The conjecture is still open. Its equivalent directed
version states that for every given tournament there exists a constant
such that every -free tournament contains a transitive
subtournament of order at least . We prove in this paper that
-free tournaments contain transitive subtournaments of
size at least for some and several
pairs of tournaments: , . In particular we prove that
-freeness implies existence of the polynomial-size transitive
subtournaments for several tournaments for which the conjecture is still
open ( stands for the \textit{complement of }). To the best of our
knowledge these are first nontrivial results of this type
Graphs with few 3-cliques and 3-anticliques are 3-universal
For given integers k, l we ask whether every large graph with a sufficiently
small number of k-cliques and k-anticliques must contain an induced copy of
every l-vertex graph. Here we prove this claim for k=l=3 with a sharp bound. A
similar phenomenon is established as well for tournaments with k=l=4.Comment: 12 pages, 1 figur
Hypergraph Ramsey numbers
The Ramsey number r_k(s,n) is the minimum N such that every red-blue coloring
of the k-tuples of an N-element set contains either a red set of size s or a
blue set of size n, where a set is called red (blue) if all k-tuples from this
set are red (blue). In this paper we obtain new estimates for several basic
hypergraph Ramsey problems. We give a new upper bound for r_k(s,n) for k \geq 3
and s fixed. In particular, we show that r_3(s,n) \leq 2^{n^{s-2}\log n}, which
improves by a factor of n^{s-2}/ polylog n the exponent of the previous upper
bound of Erdos and Rado from 1952. We also obtain a new lower bound for these
numbers, showing that there are constants c_1,c_2>0 such that r_3(s,n) \geq
2^{c_1 sn \log (n/s)} for all 4 \leq s \leq c_2n. When s is a constant, it
gives the first superexponential lower bound for r_3(s,n), answering an open
question posed by Erdos and Hajnal in 1972. Next, we consider the 3-color
Ramsey number r_3(n,n,n), which is the minimum N such that every 3-coloring of
the triples of an N-element set contains a monochromatic set of size n.
Improving another old result of Erdos and Hajnal, we show that r_3(n,n,n) \geq
2^{n^{c \log n}}. Finally, we make some progress on related hypergraph
Ramsey-type problems
Transitive Triangle Tilings in Oriented Graphs
In this paper, we prove an analogue of Corr\'adi and Hajnal's classical
theorem. There exists such that for every when the following holds. If is an oriented graph on vertices and every
vertex has both indegree and outdegree at least , then contains a
perfect transitive triangle tiling, which is a collection of vertex-disjoint
transitive triangles covering every vertex of . This result is best
possible, as, for every , there exists an oriented graph
on vertices without a perfect transitive triangle tiling in which every
vertex has both indegree and outdegree at least Comment: To appear in Journal of Combinatorial Theory, Series B (JCTB
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