A local density condition for triangles

AbstractLet G be a graph on n vertices and let α and β be real numbers, 0 < α, β < 1. Further, let G satisfy the condition that each ⌊αn⌋ subset of its vertex set spans at least βn2 edges. The following question is considered. For a fixed α what is the smallest value of β such that G contains a triangle

Extremal theory and bipartite graph-tree Ramsey numbers

AbstractFor a positive integer n and graph B, fB(n) is the least integer m such that any graph of order n and minimal degree m has a copy of B. It will be show that if B is a bipartite graph with parts of order k and l (k⩽l), then there exists a positive constant c, such that for any tree Tn of order n and for any j (0⩽j⩽(k-1)), the Ramsey number r(Tn,B)⩽n+c·(fB(n))j/(k-1) if Δ(Tn)⩽(n/(k-j-1))-(j+2)·fB(n). In particular, this implies r(Tn, B) is bounde d above by n+o(n) for any tree Tn (since fB(n)=o(n) when B is a Bipartite graph), and by n+O(1) if the tree Tn has no vertex of large degree. For special classes of bipartite graphs, such as even cycles, sharper bounds will be proved along with examples demonstrating their sharpness. Also, applications of this to the determination of Ramsey number for arbitrary graphs and trees will be discussed

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

A note on anti-coordination and social interactions

This note confirms a conjecture of [Bramoull\'{e}, Anti-coordination and social interactions, Games and Economic Behavior, 58, 2007: 30-49]. The problem, which we name the maximum independent cut problem, is a restricted version of the MAX-CUT problem, requiring one side of the cut to be an independent set. We show that the maximum independent cut problem does not admit any polynomial time algorithm with approximation ratio better than $n^{1-\epsilon}$, where $n$ is the number of nodes, and $\epsilon$ arbitrarily small, unless P=NP. For the rather special case where each node has a degree of at most four, the problem is still MAXSNP-hard.Comment: 7 page

The history of degenerate (bipartite) extremal graph problems

This paper is a survey on Extremal Graph Theory, primarily focusing on the case when one of the excluded graphs is bipartite. On one hand we give an introduction to this field and also describe many important results, methods, problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version of our survey presented in Erdos 100. In this version 2 only a citation was complete

Extremal problems involving vertices and edges on odd cycles

AbstractWe investigate the minimum, taken over all graphs G with n vertices and at least ⌊n2/4⌋ + 1 edges, of the number of vertices and edges of G which are on cycles of length 2k + 1