Goal
• Present and prove the combinatorial Ky Fan theorem
generalizes Sperner, Brouwer, Borsuk-Ulam, covering Ky Fan, ...
• Present Matoušek’s proof of Lovasz-Kneser theorem
• Present Chen’s proof of circular chromatic number of
Kneser graphs
• Quick presentation of Kneser hypergraphs
Notes and slides on
http://cermics.enpc.fr/∼meuniefr/events.html
Kneser graphs
{1, 2}
{4, 5}
{3, 4}
{3, 5}
KG(5, 2) =
{1, 3}
{2, 5}
{2, 4}
{1, 4}
m, ` two integers s.t. m ≥ 2`.
{2, 3}
Kneser graph KG(m, `):
V (KG(m, `)) =
[m]
`
n
E(KG(m, `))) = AB : A, B ∈
[m]
` ,
o
A∩B =∅
{1, 5}
Lovász-Kneser theorem
Theorem
χ(KG(m, `)) = m − 2` + 2.
First use of algebraic topology in combinatorics (1978).
All known proof are topological.
Matoušek proposed in 2003 a combinatorial (yet still
topological) proof.
Circular chromatic number
Let G = (V , E) be a graph.
(p, q)-coloring: mapping c : V → [p] s.t.
q ≤ |c(u) − c(v )| ≤ p − q
for every uv ∈ E,
(p ≥ q ≥ 1 are two integers.)
Circular chromatic number:
χc (G) = inf{p/q : G admits a (p, q)-coloring}.
Known fact:
χ(G) − 1 < χc (G) ≤ χ(G).
Circular chromatic number of Kneser graphs
Theorem (Chen 2011)
χc (KG(m, `)) = χ(KG(m, `))
for even m, already proved by Simonyi, Tardos, and M..
Chen’s proof is combinatorial: extension of Matoušek’s proof.
No “continuous” proof known.
Kneser hypergraphs
m, `, r three integers s.t. m ≥ r `.
Kneser hypergraph KGr (m, `):
V (KGr (m, `)) =
[m]
`
n
E(KGr (m, `)) = {A1 , . . . , Ar } : Ai ∈
[m]
` ,
Ai ∩ Aj = ∅ for i 6= j
o
Chromatic number
Theorem (Alon-Frankl-Lovász theorem)
m − r (` − 1)
χ(KG (m, `)) =
r −1
r
Can also be proved via a combinatorial (yet topological)
approach, Ziegler 2003.
This approach can be used to prove other results
(“local chromatic number”, Hedetniemi’s conjecture for hypergraphs,...).