Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Multigraphs (only) satisfy a weak triangle removal lemma. | Multigraphs only satisfy a weak triangle removal lemma Asaf Shapira Raphael Yusteh Submitted Dec 30 2008 Accepted Mar 5 2009 Published Mar 20 2009 Mathematics Subject Classification 05C35 Abstract The triangle removal lemma states that a simple graph with o n3 triangles can be made triangle-free by removing o n2 edges. It is natural to ask if this widely used result can be extended to multi-graphs. In this short paper we rule out the possibility of such an extension by showing that there are multi-graphs with only n2 o 1 triangles that are still far from being triangle-free. On the other hand we show that for some slowly growing function g n w 1 if a multi-graph has only g n n2 triangles then it must be close to being triangle-free. The proof relies on variants of the Ruzsa-Szemeredi theorem 15 . 1 Introduction Motivated by a problem in the theory of extremal hypergraphs Ruzsa and Szemeredi 15 proved the following two theorems. Theorem 1 Ruzsa and Szemeredi 15 If G is an n vertex graph from which one should remove at least en2 edges in order to destroy all triangles then G contains at least f e n3 triangles where f e 0. Theorem 2 Ruzsa and Szemeredi 15 Suppose S c n is a set of integers containing no 3-term arithmetic progression. Then there is a graph G V E with V 6n and E 3n S whose edges can be uniquely partitioned into n S edge disjoint triangles. Furthermore G contains no other triangles. School of Mathematics and College of Computing Georgia Institute of Technology Atlanta GA 30332. E-mail asafico@ Department of Mathematics University of Haifa Haifa 31905 Israel. E-mail raphy@ THE ELECTRONIC JOURNAL OF COMBINATORICS 16 2009 N11 1 These two theorems turned out to be two of the most influential results in extremal combinatorics. First a simple application of these two theorems gives a short proof of Roth s Theorem 14 stating that a subset of n of size en contains a 3-term arithmetic progression. The results in 15 were followed by
