Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Induced trees in triangle-free graphs. | Induced trees in triangle-free graphs Jiri Matousek Robert Samal Department of Applied Mathematics and Institute of Theoretical Computer Science ITI Charles University Malostranské nám. 25 118 00 Praha 1 Czech Republic Submitted Nov 29 2007 Accepted Feb 24 2008 Published Mar 12 2008 Mathematics Subject Classification 05C55 05C05 Abstract We prove that every connected triangle-free graph on n vertices contains an induced tree on exp cựlogn vertices where c is a positive constant. The best known upper bound is 2 o 1 pn. This partially answers questions of Erdos Saks and Sos and of Pultr. 1 Introduction For a graph G let t G denote the maximum number of vertices of an induced subgraph of G that is a tree . connected and acyclic . There are arbitrary large graphs G with t G 2 namely graphs in which every connected component is a clique. To rule out these trivial examples we need to put some restrictions on G. Motivated by study of forbidden configurations in Priestley spaces 1 Pultr private communication 2002 asked how big t G can be if G is connected and bipartite. Formally he was interested about asymptotic properties of the function fB n min t G V G n G connected and bipartite . Pultr s question was the starting point of our work. However the function t G was studied earlier and in a more general context by Erdos Saks and Sós 2 . They describe the influence of the number of edges of G on t G and more to our point they study how small t G can be if G is given. They observe that t G 2a G and this allows Currently on leave from Institute for Theoretical Computer Science ITI . The paper was finished while the second author was a PIMS postdoctoral fellow at Department of Mathematics Simon Fraser University Burnaby . V5A 1S6 Canada. THE ELECTRONIC JOURNAL OF COMBINATORICS 15 2008 R41 1 them to use estimates for Ramsey numbers. This way they show that for any fixed k 3 there are constants C1 c2 such that C1 gn min t G V G n G 2 Kk C2 log n. log log n For k 3 the .