Independence densities of hypergraphs

1. 2014 CMS Summer Meeting Independence densities of hypergraphs Anthony Bonato Ryerson University Independence densities - Anthony Bonato

2. Paths • number of independent sets • = F(n+2) • Fibonacci number … Pn Independence densities - Anthony Bonato

3. Stars • number of independent sets = 1+2n • independence density • = 2-n-1+½ … K1,n Independence densities - Anthony Bonato

4. Independence density • G order n • i(G) = number of independent sets in G (including ∅) • Fibonacci number of G • id(G) = i(G) / 2n • independence density of G Independence densities - Anthony Bonato

5. Properties • if G is a spanning subgraph of H, then i(H) ≤ i(G) • i(G U H) = i(G)i(H) • if G is subgraph of H, then id(H) ≤ id(G) • G has an edge, then: id(G) ≤ id(K2) = 3/4 • id(G U H) = id(G)id(H) Independence densities - Anthony Bonato

6. Infinite graphs? • view countably a infinite graph as a limit of chains • extend definition by continuity: • well-defined? • possible values? Independence densities - Anthony Bonato

7. Chains • let G be a countably infinite graph • a chainCin G is a set of induced subgraphs Gi such that: • for all i, Giis an induced subgraph of Gi+1and • write id(G, C) = Independence densities - Anthony Bonato

8. Existence and uniqueness Theorem (B,Brown,Kemkes,Pralat,11) Let G be a countably infinite graph. • For each chain C, id(G, C) exists. • For all chains C and C’ in G, id(G, C)=id(G, C’). Independence densities - Anthony Bonato

9. Examples • stars: id(K1,∞) = 1/2 • one-way infinite path: id(P∞) = 0 Independence densities - Anthony Bonato

10. Bounds on id • ifGcontains and infinite matching, then id(G) = 0 • matching number of G, written µ(G), is the supremum of the cardinalities of pairwise non-intersecting edges in G Theorem (B,Brown,Kemkes,Pralat,14) If µ(G) is finite, then: • in particular, id(G) = 0 iffµ(G) is infinite Independence densities - Anthony Bonato

11. Rationality Theorem(BBKP,11) Let G be a countable graph. • id(G) is rational. • The closure of the set {id(G): G countable} is a subset of the rationals. Independence densities - Anthony Bonato

12. Aside: other densities • many other density notions for graphs and hypergraphs: • upper density • homomorphism density • Turán density • co-degree density • cop density, … Independence densities - Anthony Bonato

13. Question • hereditary graph class X: closed under induced subgraphs • egs: X = independent sets; cliques; triangle-free graphs; perfect graphs; H-free graphs • Xd(G) = proportion of subsets which induce a graph in X • generalizes to infinite graphs via chains • Is Xd(G) rational? Independence densities - Anthony Bonato

14. Hypergraphs • hypergraphH = (V,E), E = hyperedges • independent set: does not contain a hyperedge • id(H) defined analogously • extend to infinite hypergraphs by continuity • well-defined Independence densities - Anthony Bonato

15. 1 2 3 4 Examples ∅,{1},{2},{3},{4}, {1,2},{1,3},{2,3}, {1,4},{3,4}, {1,3,4} id(H) = 11/16 H

16. Examples, cont id(H) = 7/8 … Independence densities - Anthony Bonato

17. Hypergraph id’s examples: • graph, E= subsets of vertices containing a copy of K2 • recovers the independence density of graphs • graph, fix a finite graph F; E = subsets of vertices containing a copy of F • F-free density (generalizes (1)). • relational structure (graphs, digraphs, orders, etc); F a set of finite structures; E = subsets of vertices containing a member of F • F-free density of a structure (generalizes (2)) Independence densities - Anthony Bonato

18. Bounds on id • matching number of hypergraph H, written µ(H), is the supremum of the cardinalities of pairwise non-intersecting hyperedges in H Theorem (B,Brown,Mitsche,Pralat,14) Let H be a hypergraph whose hyperedges have cardinality bounded by k > 0. If µ(H) is finite, then: • sharp if k = 1,2; not sure lower bound is sharp if k > 2 Independence densities - Anthony Bonato

19. Rationality • rank k hypergraph: hyperedges bounded in cardinality by k > 0 • finite rank: rank k for some k Theorem (BBMP,14): If H has finite rank, then id(H) is rational. Independence densities - Anthony Bonato

20. Sketch of proof • notation: for finite disjoint sets of vertices A and B idA,B(H) = density of independent sets containing A and not B • analogous properties to id(H) = id∅,∅(H) Independence densities - Anthony Bonato

21. Properties of idA,B(H) • For a vertex x outside A U B: . • For a set W outside A U B: Independence densities - Anthony Bonato

22. Out-sets • for a given A, B, and any hyperedge S such that S∩B = ∅, the set S \ A is the out-set of S relative to A and B • example: AB S • notation: idrA,B(H) denotes that every out-set has cardinality at most r • note that: idk∅,∅(H) = id(H) Independence densities - Anthony Bonato

23. Claims • If A is not independent, then idrA,B(H) = 0. • If A is independent and there is an infinite family of disjoint out-sets, then idrA,B(H) = 0. • If A is independent, then id0A,B(H)=2-(|A|+|B|). • Suppose that A is independent and there is no infinite family of disjoint out-sets. If O1, O2,…, Osis a maximal family of disjoint out-sets, then for all r > 0, where W is the union of the Oi. Independence densities - Anthony Bonato

24. Final steps… • start with idk∅,∅(H) • by (2), assume wlog there are finitely many out-sets • as A and B are empty, the out-sets are disjoint hyperedges with union W • by (4): • apply induction using (1)-(3); each term is rational, or a finite sum of terms where we can apply (1)-(3) • process ends after k steps Independence densities - Anthony Bonato

25. Unbounded rank H … • Theorem (BBMP,14) • The independence density of H is • . • 2. The value is S is irrational. • proof of (2) uses Euler’s Pentagonal Number Theorem Independence densities - Anthony Bonato

26. Any real number • case of finite, but unbounded hyperedges • Hunb= {x: there is a countable hypergraph H with id(H) = x} Theorem (BBMP,14) Hunb = [0,1]. • contrasts with rank k case, where there exist gaps such as (1-1/2k,1) Independence densities - Anthony Bonato

27. Independence polynomials • H finite, independence polynomial of Hwrtx > 0 i(H,x) = • for eg: id(H) = i(H,1)/2n • example • i(Pn,x) = i(Pn-1,x) + xi(Pn-2,x), i(P1,x)=1+x, (P2,x)=1+2x Independence densities - Anthony Bonato

28. Independence densities at x • (,x): defined in natural way for fixed x ≥ 0 • may depend on chain if x ≠ 1 • (,1) = id(H) • examples: • (,x) = 0 for all x • generalizes to chains with βn= o(n) • (H,x) = 0 if x < 1: • bounded above by ((1+x)/2)n=o(1) Independence densities - Anthony Bonato

29. Examples, continued • + • with chain (Pn: n ≥ 1), we derive that: (,x) = Independence densities - Anthony Bonato

30. Examples, continued • for each r > 1, can choose chain C such that ) = • r is a jumping point Independence densities - Anthony Bonato

31. Future directions • classify gaps among densities for given hypergraphs • rationality of closure of set of id’s for rank k hypergraphs • which hypergraphs have jumping points, and what are their values? Independence densities - Anthony Bonato

32. General densities • d: hypergraph function satisfying: • multiplicative on disjoint unions • monotone increasing on subgraphs • d(H) well-defined for infinite hypergraphs • properties of d(H)? for eg, when rational? Independence densities - Anthony Bonato