58,264 research outputs found
Book Ramsey numbers I
A book of size q is the union of q triangles sharing a common edge. We find
the exact Ramsey number of books of size q versus books of size p when
p<q/6-o(q).Comment: 21 pages. Submitte
Extremális Problémák Diszkrét Struktúrákban = Extremal Problems in Discrete Structures
A négy éves projekt során 17 cikkünk jelent meg, 2 nyomdában van, 2 elfogadott és 2 benyújtott. Örvendetes, hogy a pályázókon kivül sikerült 17 további kutatót (fiatalokat és szeniorokat egyaránt) megnyerni témáinknak, melyeket - némi önkénnyel - igy csoportositok (a számok a közlemények sorszámai a zárójelentésből): 1. Ramsey elmélet. Kiemelem a regularitás lemma új alkalmazásait, melyek a terület előtérben lévő kutatásaihoz tartoznak, pl. 3,4,8,9,10,16,17 - legtöbbjükben társszerző Szemerédi Endre is. Jelentős cikknek tartom a 6. könyvfejezetet is, mely egy utóbbi időben egyre aktivabb területet foglal össze. 2. Gallai szinezések. Ez valójában a Ramsey elmélet ás extremális kombinatorika határán van, izgalmas, de nehéz kérdéseket vet fel Gallai egy alapvető technikájának általánositási lehetőségeiről, 1,11,18. 3. Kódok és extremális problémák. Kiemelendő Füredi és Ruszinkó 23-as igen tartalmas és gondosan megirt cikke, melyet az Advances in Mathematicshoz nyujtottak be és 7. cikk, mely gráfok perfektségének alapvető mértékszámát kapcsolja össze a Ramsey gráfok elméletével. Néhány további eredményünk (14, 19,20,21) nem kapcsolódik alapvető témákhoz, de azokban is új eredményeket fejlesztünk tovább az extremális gráf és hipergráfelmélet területéről. | During the four year project 17 papers are published, 2 in press, 2 are accepted and 2 are submitted. We are glad that apart from the three persons participating in the project, we could attract 17 further researchers (both young and seniors) to our subjects. Somewhat arbitrarily, these subjects can be grouped as follows (the numbers refer to the numbering of the list of publications). 1.Ramsey theory. I pinpoint some new applications of the Regularity Lemma related to recent research areas, namely 3,4,8,9,10,16,17 - most with coauthor Endre Szemerédi. I think that 6., a book chapter is also significant, reviewing an area with increasing of present activity. 2. Gallai colorings. This area is on the border of Ramsey theory and extremal combinatorics, posing exciting but difficult questions on possible extensions of a basic technique of Gallai, 1,11,18. 3. Codes and extremal problems. Here I emphasize 23, a deep and carefully written paper of Füredi and Ruszinkó submitted to Advances in Mathematics and 7, which relates a basic measure of perfectness of graphs to the theory of Ramsey graphs. Some of our further results (14,19,20,21) does not relate to basic results but still extends recent results of extremal graph and hypergraph theory
Ramsey Goodness and Beyond
In a seminal paper from 1983, Burr and Erdos started the systematic study of
Ramsey numbers of cliques vs. large sparse graphs, raising a number of
problems. In this paper we develop a new approach to such Ramsey problems using
a mix of the Szemeredi regularity lemma, embedding of sparse graphs, Turan type
stability, and other structural results. We give exact Ramsey numbers for
various classes of graphs, solving all but one of the Burr-Erdos problems.Comment: A new reference is adde
A note on Ramsey Numbers for Books
A book of size N is the union of N triangles sharing a common edge. We show
that the Ramsey number of a book of size N vs. a book of size M equals 2N+3 for
all N>(10^6)M. Our proof is based on counting.Comment: 9 pages, submitted to Journal of Graph Theory in Aug 200
Large generalized books are p-good
An r-book of size q is a union of q (r+1)-cliques sharing a common r-clique.
We find exactly the Ramsey number of a p-clique versus r-books of sufficiently
large size. Furthermore, we find asymptotically the Ramsey number of any fixed
p-chromatic graph versus r-books of sufficiently large size. The key element in
our proofs is Szemeredi's Regularity Lemma.Comment: 16 pages, accepted in JCT
On a problem of Erd\H{o}s and Rothschild on edges in triangles
Erd\H{o}s and Rothschild asked to estimate the maximum number, denoted by
H(N,C), such that every N-vertex graph with at least CN^2 edges, each of which
is contained in at least one triangle, must contain an edge that is in at least
H(N,C) triangles. In particular, Erd\H{o}s asked in 1987 to determine whether
for every C>0 there is \epsilon >0 such that H(N,C) > N^\epsilon, for all
sufficiently large N. We prove that H(N,C) = N^{O(1/log log N)} for every fixed
C < 1/4. This gives a negative answer to the question of Erd\H{o}s, and is best
possible in terms of the range for C, as it is known that every N-vertex graph
with more than (N^2)/4 edges contains an edge that is in at least N/6
triangles.Comment: 8 page
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
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