Triangle-intersecting Families of Graphs

A family of graphs F is triangle-intersecting if for every G,H 2 F, G\H contains a triangle. A conjecture of Simonovits and Sos from 1976 states that the largest triangle-intersecting families of graphs on a fixed set of n vertices are those obtained by fixing a specific triangle and taking all graphs containing it, resulting in a family of size 1 2 ( n 2) . We prove this conjecture and some generalizations (for example, we prove that the same is true of odd-cycle-intersecting families, and we obtain best possible bounds on the size of the family under dierent, not necessarily uniform, measures). We also obtain stability results, showing that almost-largest triangle-intersecting families have approximately the same structure.

[1]  A. J. Hoffman,et al.  ON EIGENVALUES AND COLORINGS OF GRAPHS, II , 1970 .

[2]  Noam Nisan,et al.  On the degree of boolean functions as real polynomials , 1992, STOC '92.

[3]  Haran Pilpel,et al.  Intersecting Families of Permutations , 2010, 1011.3342.

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

[5]  Fan Chung Graham,et al.  Some intersection theorems for ordered sets and graphs , 1986, J. Comb. Theory, Ser. A.

[6]  Paul A. Russell.  Families intersecting on an interval , 2009, Discret. Math..

[7]  P. McAree,et al.  Using Sturm sequences to bracket real roots of polynomial equations , 1990 .

[8]  Zoltán Füredi,et al.  Representations of families of triples over GF(2) , 1990, J. Comb. Theory, Ser. A.

[9]  P. Erdös,et al.  INTERSECTION THEOREMS FOR SYSTEMS OF FINITE SETS , 1961 .

[10]  Ehud Friedgut,et al.  On the measure of intersecting families, uniqueness and stability , 2008, Comb..

[11]  Jerrold R. Griggs,et al.  Anticlusters and intersecting families of subsets , 1989, J. Comb. Theory, Ser. A.

[12]  S. Safra,et al.  Noise-Resistant Boolean-Functions are Juntas , 2003 .

[13]  Rudolf Ahlswede,et al.  The Complete Intersection Theorem for Systems of Finite Sets , 1997, Eur. J. Comb..

[14]  V. Rich.  Personal communication , 1989, Nature.

[15]  Norihide Tokushige,et al.  The exact bound in the Erdös-Ko-Rado theorem for cross-intersecting families , 1989, J. Comb. Theory, Ser. A.

[16]  Richard M. Wilson,et al.  The exact bound in the Erdös-Ko-Rado theorem , 1984, Comb..