Ramsey graphs induce subgraphs of quadratically many sizes

An n-vertex graph is called C-Ramsey if it has no clique or independent set of size C log n. All known constructions of Ramsey graphs involve randomness in an essential way, and there is an ongoing line of research towards showing that in fact all Ramsey graphs must obey certain "richness" properties characteristic of random graphs. Motivated by an old problem of Erd\H{o}s and McKay, recently Narayanan, Sahasrabudhe and Tomon conjectured that for any fixed C, every n-vertex C-Ramsey graph induces subgraphs of $\Theta(n^2)$ different sizes. In this paper we prove this conjecture.

[1]  Noga Alon,et al.  Induced subgraphs with distinct sizes , 2009, Random Struct. Algorithms.

[2]  P. Erdos,et al.  On a Ramsey type theorem , 1972 .

[3]  Noga Alon,et al.  Induced subgraphs of prescribed size , 2003 .

[4]  G. Szekeres,et al.  A combinatorial problem in geometry , 2009 .

[5]  D. Zuckerman,et al.  Explicit two-source extractors and resilient functions , 2016, Electron. Colloquium Comput. Complex..

[6]  Wojciech Samotij,et al.  Sizes of Induced Subgraphs of Ramsey Graphs , 2009, Combinatorics, Probability and Computing.

[7]  Avi Wigderson,et al.  2-source dispersers for sub-polynomial entropy and Ramsey graphs beating the Frankl-Wilson construction , 2006, STOC '06.

[8]  Saharon Shelah,et al.  Erdos and Rényi Conjecture , 1998, J. Comb. Theory, Ser. A.

[9]  Kevin P. Costello.  Bilinear and quadratic variants on the Littlewood-Offord problem , 2009 .

[10]  Paul Erdős.  Some recent problems and results in graph theory , 1997 .

[11]  Bhargav P. Narayanan,et al.  Ramsey Graphs Induce Subgraphs of Many Different Sizes , 2019, Comb..

[12]  Jean Bourgain,et al.  On the singularity probability of discrete random matrices , 2009, 0905.0461.

[13]  Maria Axenovich,et al.  Graphs Having Small Number of Sizes on Induced k-Subgraphs , 2007, SIAM J. Discret. Math..

[14]  Alan M. Frieze,et al.  On Subgraph Sizes in Random Graphs , 1992, Comb. Probab. Comput..

[15]  Bhargav P. Narayanan,et al.  Induced Subgraphs With Many Distinct Degrees , 2018, Comb. Probab. Comput..

[16]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[17]  P. Erdös.  Some remarks on the theory of graphs , 1947 .

[18]  Vojtech Rödl,et al.  Non-Ramsey Graphs Are c log n-Universal , 1999, J. Comb. Theory, Ser. A.

[19]  Benny Sudakov,et al.  Induced subgraphs of Ramsey graphs with many distinct degrees , 2007, J. Comb. Theory, Ser. B.

[20]  Peter Frankl,et al.  Intersection theorems with geometric consequences , 1981, Comb..

[21]  Gil Cohen,et al.  Two-source dispersers for polylogarithmic entropy and improved ramsey graphs , 2015, Electron. Colloquium Comput. Complex..

[22]  Noga Alon,et al.  Graphs with a small number of distinct induced subgraphs , 1989, Discret. Math..

[23]  A. Hajnal.  On spanned subgraphs of graphs , 1977 .

[24]  P. Erdös.  On Some of my Favourite Problems in Various Branches of Combinatorics , 1992 .

[25]  Avi Wigderson,et al.  2-source dispersers for $n^{o(1)}$ entropy, and Ramsey graphs beating the Frankl-Wilson construction , 2012 .