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

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

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

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

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(K 2 ) = 3/4 id(G U H) = id(G)id(H) Independence densities - Anthony Bonato 5

Infinite graphs? Independence densities - Anthony Bonato 6

Chains Independence densities - Anthony Bonato 7

Existence and uniqueness Theorem (B,Brown,Kemkes,Pralat,11) Let G be a countably infinite graph. 1.For each chain C, id(G, C ) exists. 2.For all chains C and C ’ in G, id(G, C )=id(G, C ’). Independence densities - Anthony Bonato 8

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

Bounds on id Independence densities - Anthony Bonato 10

Rationality Theorem (BBKP,11) Let G be a countable graph. 1.id(G) is rational. 2.The closure of the set {id(G): G countable} is a subset of the rationals. Independence densities - Anthony Bonato 11

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 12

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 13

Hypergraphs hypergraph H = (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 14

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

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

Hypergraph id’s examples: 1.graph, E = subsets of vertices containing a copy of K 2 –recovers the independence density of graphs 2.graph, fix a finite graph F; E = subsets of vertices containing a copy of F –F-free density (generalizes (1)). 3.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 17

Bounds on id Independence densities - Anthony Bonato 18

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 19

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

Properties of id A,B (H) Independence densities - Anthony Bonato 21

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: A B S notation: id r A,B (H) denotes that every out-set has cardinality at most r note that: id k ∅, ∅ (H) = id(H) Independence densities - Anthony Bonato 22

Claims Independence densities - Anthony Bonato 23

Final steps… Independence densities - Anthony Bonato 24

Unbounded rank Independence densities - Anthony Bonato 25 … H

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

Independence polynomials Independence densities - Anthony Bonato 27

Independence densities at x Independence densities - Anthony Bonato 28

Examples, continued Independence densities - Anthony Bonato 29

Examples, continued Independence densities - Anthony Bonato 30

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 31

General densities Independence densities - Anthony Bonato 32