370 research outputs found
On two problems in Ramsey-Tur\'an theory
Alon, Balogh, Keevash and Sudakov proved that the -partite Tur\'an
graph maximizes the number of distinct -edge-colorings with no monochromatic
for all fixed and , among all -vertex graphs. In this
paper, we determine this function asymptotically for among -vertex
graphs with sub-linear independence number. Somewhat surprisingly, unlike
Alon-Balogh-Keevash-Sudakov's result, the extremal construction from
Ramsey-Tur\'an theory, as a natural candidate, does not maximize the number of
distinct edge-colorings with no monochromatic cliques among all graphs with
sub-linear independence number, even in the 2-colored case.
In the second problem, we determine the maximum number of triangles
asymptotically in an -vertex -free graph with . The
extremal graphs have similar structure to the extremal graphs for the classical
Ramsey-Tur\'an problem, i.e.~when the number of edges is maximized.Comment: 22 page
An approximate version of Sidorenko's conjecture
A beautiful conjecture of Erd\H{o}s-Simonovits and Sidorenko states that if H
is a bipartite graph, then the random graph with edge density p has in
expectation asymptotically the minimum number of copies of H over all graphs of
the same order and edge density. This conjecture also has an equivalent
analytic form and has connections to a broad range of topics, such as matrix
theory, Markov chains, graph limits, and quasirandomness. Here we prove the
conjecture if H has a vertex complete to the other part, and deduce an
approximate version of the conjecture for all H. Furthermore, for a large class
of bipartite graphs, we prove a stronger stability result which answers a
question of Chung, Graham, and Wilson on quasirandomness for these graphs.Comment: 12 page
The Ramsey multiplicity of K_4
With the help of computer algorithms, we improve the lower bound on the Ramsey multiplicity of K4, and thus show that the exact value of it is equal to 9
Extremal problems and results related to Gallai-colorings
A Gallai-coloring (Gallai--coloring) is an edge-coloring (with colors from
) of a complete graph without rainbow triangles. Given a
graph and a positive integer , the -colored Gallai-Ramsey number
is the minimum integer such that every Gallai--coloring of the
complete graph contains a monochromatic copy of . In this paper, we
prove that for any positive integers and , there exists a constant
such that if is an -vertex graph with maximum degree , then
is at most . We also determine for the graph on 5 vertices
consisting of a with a pendant edge. Furthermore, we consider two
extremal problems related to Gallai--colorings. For , we
determine upper and lower bounds for the minimum number of monochromatic
triangles in a Gallai--coloring of , implying that this number is
and yielding the exact value for . We also determine upper and
lower bounds for the maximum number of edges that are not contained in any
rainbow triangle or monochromatic triangle in a -edge-coloring of .Comment: 20 pages, 1 figur
Ramsey multiplicity and the Tur\'an coloring
Extending an earlier conjecture of Erd\H{o}s, Burr and Rosta conjectured that
among all two-colorings of the edges of a complete graph, the uniformly random
coloring asymptotically minimizes the number of monochromatic copies of any
fixed graph . This conjecture was disproved independently by Sidorenko and
Thomason. The first author later found quantitatively stronger counterexamples,
using the Tur\'an coloring, in which one of the two colors spans a balanced
complete multipartite graph.
We prove that the Tur\'an coloring is extremal for an infinite family of
graphs, and that it is the unique extremal coloring.
This yields the first determination of the Ramsey multiplicity constant of a
graph for which the Burr--Rosta conjecture fails.
We also prove an analogous three-color result. In this case, our result is
conditional on a certain natural conjecture on the behavior of two-color Ramsey
numbers.Comment: 37 page
Tur\'an Colourings in Off-Diagonal Ramsey Multiplicity
The Ramsey multiplicity constant of a graph is the limit as tends to
infinity of the minimum density of monochromatic labelled copies of in a
colouring of the edges of with two colours. Fox and Wigderson recently
identified a large family of graphs whose Ramsey multiplicity constants are
attained by sequences of "Tur\'an colourings;" i.e. colourings in which one of
the colour classes forms the edge set of a balanced complete multipartite
graph. The graphs in their family come from taking a connected non-3-colourable
graph with a critical edge and adding many pendant edges. We extend their
result to an off-diagonal variant of the Ramsey multiplicity constant which
involves minimizing a weighted sum of red copies of one graph and blue copies
of another. We also apply the flag algebra method to investigate the minimum
number of pendant edges required for Tur\'an colourings to become optimal when
the underlying graphs are small cliques.Comment: 48 pages, 2 figure
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