Abstract
A family F of graphs is triangle-intersecting if for every G;H ε F, G ∩ H contains a triangle. A conjecture of Simonovits and Sós 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 1/8 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 different, not necessarily uniform, measures). We also obtain stability results, showing that almost-largest triangle-intersecting families have approximately the same structure.
Original language | English |
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Pages (from-to) | 841-885 |
Number of pages | 45 |
Journal | Journal of the European Mathematical Society |
Volume | 14 |
Issue number | 3 |
DOIs | |
State | Published - 2012 |
Externally published | Yes |
Keywords
- Discrete Fourier analysis
- Graphs
- Intersecting families
ASJC Scopus subject areas
- General Mathematics
- Applied Mathematics