Bipartite induced density in triangle-free graphs

Any triangle-free graph on $n$ vertices with minimum degree at least $d$ contains a bipartite induced subgraph of minimum degree at least $d^2/(2n)$. This is sharp up to a logarithmic factor in $n$. We also provide a related extremal result for the fractional chromatic number.

[1]  László Lovász,et al.  On the ratio of optimal integral and fractional covers , 1975, Discret. Math..

[2]  Alan M. Frieze,et al.  Random graphs , 2006, SODA '06.

[3]  Gonzalo Fiz Pontiveros,et al.  The triangle-free process and R(3,k) , 2013 .

[4]  Joel H. Spencer,et al.  Asymptotic lower bounds for Ramsey functions , 1977, Discret. Math..

[5]  Paul Erdös,et al.  Chromatic number of finite and infinite graphs and hypergraphs , 1985, Discret. Math..

[6]  János Komlós,et al.  A Note on Ramsey Numbers , 1980, J. Comb. Theory, Ser. A.

[7]  W. T. Gowers,et al.  RANDOM GRAPHS (Wiley Interscience Series in Discrete Mathematics and Optimization) , 2001 .

[8]  Béla Bollobás,et al.  Random Graphs , 1985 .

[9]  Robert Morris,et al.  The Triangle-Free Process and the Ramsey Number 𝑅(3,𝑘) , 2020 .

[10]  J. Pach,et al.  Wiley‐Interscience Series in Discrete Mathematics and Optimization , 2011 .

[11]  S. Brandt Dense triangle-free graphs are four-colorable : A solution to the Erdős-Simonovits problem , 2005 .

[12]  Luke Postle,et al.  Fractional coloring with local demands , 2018, ArXiv.

[13]  D. de Werra,et al.  Graph Coloring Problems , 2013 .

[14]  Zsolt Tuza,et al.  Bipartite Subgraphs of Triangle-Free Graphs , 1994, SIAM J. Discret. Math..

[15]  Jeong Han Kim,et al.  The Ramsey Number R(3, t) Has Order of Magnitude t2/log t , 1995, Random Struct. Algorithms.

[16]  Rémi de Joannis de Verclos,et al.  Coloring triangle‐free graphs with local list sizes , 2020, Random Struct. Algorithms.

[17]  Stéphan Thomassé,et al.  Separation Choosability and Dense Bipartite Induced Subgraphs , 2018, Combinatorics, Probability and Computing.

[18]  Carsten Thomassen,et al.  Coloring triangle-free graphs with fixed size , 2000, Discret. Math..

[19]  Benny Sudakov,et al.  Dense Induced Bipartite Subgraphs in Triangle-Free Graphs , 2018, Comb..

[20]  Noga Alon,et al.  Choice Numbers of Graphs: a Probabilistic Approach , 1992, Combinatorics, Probability and Computing.

[21]  Noga Alon Triangle-free graphs with large chromatic numbers , 2000, Discret. Math..

[22]  Rémi de Joannis de Verclos,et al.  Occupancy fraction, fractional colouring, and triangle fraction , 2018, ArXiv.

[23]  A. Leaf GRAPH THEORY AND PROBABILITY , 1957 .

[24]  János Komlós,et al.  A Dense Infinite Sidon Sequence , 1981, Eur. J. Comb..

[25]  James B. Shearer,et al.  A note on the independence number of triangle-free graphs , 1983, Discret. Math..

[26]  Carsten Thomassen,et al.  On the Chromatic Number of Triangle-Free Graphs of Large Minimum Degree , 2002, Comb..

[27]  Miklós Simonovits,et al.  On a valence problem in extremal graph theory , 1973, Discret. Math..

[28]  Tom Bohman,et al.  Dynamic concentration of the triangle‐free process , 2013, Random Struct. Algorithms.

[29]  Michael Molloy,et al.  The list chromatic number of graphs with small clique number , 2017, J. Comb. Theory B.