1 Don’t be dense, try hypergraphs! Anthony Bonato Ryerson University Ryerson

Independent sets set of vertices in an undirected, simple graph, no two of which are adjacent 2

Paths 3 … number of independent sets = F(n+2) -Fibonacci number PnPn

Stars 4 number of independent sets = 1+2 n independence density = 2 -n-1 +½ K 1,n …

Independence density G a graph of 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 –rational in (0,1] 5

Properties if G is a spanning subgraph of H, then i(H) ≤ i(G) –(possibly) fewer edges in G i(G U H) = i(G)i(H) 6

Monotonicity if G is subgraph of H, then id(H) ≤ id(G) –G has an edge, then: id(G) ≤ id(K 2 ) = ¾ Proof: Say G has order m and H has order n. id(H)= i(H) / 2 n ≤ i(G U (H-G)) / 2 n (disjoint union) = i(G)i(H-G) / 2 m 2 n-m = id(G)id(H-G) ≤ id(G). id(G U H) = id(G)id(H) 7

Infinite graphs? 8

Chains 9

Existence and uniqueness 10

Examples infinite star: id(K 1,∞ ) = 1/2 one-way infinite path: id(P ∞ ) = 0 11

Bounds on id 12

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. 13

Aside: other densities many other density notions for graphs and hypergraphs: –upper density –homomorphism density –Turán density –co-degree density –cop density, … 14

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? 15

Try 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 16

Examples ∅,{1},{2},{3},{4}, {1,2},{1,3},{2,3}, {1,4},{3,4}, {1,3,4} id(H) = 11/16 H 17

Examples, cont 18 … 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)) 19

Bounds on id 20

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. 21

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) 22

Properties of id A,B (H) 23

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) 24

Claims 25

Final steps… 26

Unbounded rank 27 … 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) 28

Independence polynomials 29

Independence densities at x 30

Examples, continued 31

Asymptotic behaviour 32

Sketch of proof 33

Examples, continued 34

Future directions classify gaps among densities for given hypergraphs –eg: rank k, (1-1/2 k,1) is a gap rationality of closure of set of id’s for rank k hypergraphs which hypergraphs have jumping points, and what are their values? 35

General densities 36