An Erdős-Ko-Rado theorem for unions of length 2 paths

A family of sets is intersecting if any two sets in the family intersect. Given a graph G and an integer r ≥ 1, let I(r)(G) denote the family of independent sets of size r of G. For a vertex v of G, the family of independent sets of size r that contain v is called an r-star. Then G is said to be r-EKR if no intersecting subfamily of I(r)(G) is bigger than the largest r-star. Let n be a positive integer, and let G consist of the disjoint union of n paths each of length 2. We prove that if 1 ≤ r ≤ n/2, then G is r-EKR. This affirms a longstanding conjecture of Holroyd and Talbot for this class of graphs and can be seen as an analogue of a well-known theorem on signed sets, proved using different methods, by Deza and Frankl and by Bollobás and Leader. Our main approach is a novel probabilistic extension of Katona’s elegant cycle method, which might be of independent interest.

[1]  Anthony J. W. Hilton,et al.  A generalization of Talbot's theorem about King Arthur and his Knights of the Round Table , 2009, J. Comb. Theory, Ser. A.

[2]  Gyula O. H. Katona.  A simple proof of the Erd?s-Chao Ko-Rado theorem , 1972 .

[3]  Peter Frankl,et al.  Erdös–Ko–Rado Theorem—22 Years Later , 1983 .

[4]  B. Bollobás,et al.  An Erdős-Ko-Rado Theorem For Signed Sets , 1997 .

[5]  Konrad Engel.  An Erdös-Ko-Rado theorem for the subcubes of a cube , 1984, Comb..

[6]  Jun Wang,et al.  Cross-intersecting families and primitivity of symmetric systems , 2010, J. Comb. Theory, Ser. A.

[7]  Peter Borg.  Intersecting Systems of Signed Sets , 2007, Electron. J. Comb..

[8]  Peter Borg,et al.  On the Hilton-Spencer intersection theorems for unions of cycles , 2019 .

[9]  Fred C. Holroyd,et al.  Compression and Erdös-Ko-Rado graphs , 2005, Discret. Math..

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

[11]  Rudolf Ahlswede,et al.  The Diametric Theorem in Hamming Spaces-Optimal Anticodes , 1998 .

[12]  Imre Leader,et al.  Multiple cross-intersecting families of signed sets , 2010, J. Comb. Theory, Ser. A.

[13]  Fred C. Holroyd,et al.  Graphs with the Erdös-Ko-Rado property , 2003, Discret. Math..

[14]  C. L. Spencer,et al.  A Graph-theoretical Generalization of Berge’s Analogue of the Erdős-Ko-Rado Theorem , 2006 .

[15]  Christian Bey.  An intersection theorem for weighted sets , 2001, Discret. Math..