The famous Lovász Local Lemma was derived in the paper of P. Erdős and Lovász to prove that any _n_-uniform non-_r_-colorable hypergraph _H_ has maximum edge degree at least _Δ(H) ≥ ¼ r_<sup>_n−1_</sup>. A long series of papers is devoted to the improvement of this classical result for different classesof uniform hypergraphs. In our work we deal with colorings of simple hypergraphs, i.e. hypergraphs in which everytwo distinct edges do not share more than one vertex. By using a multipass random recoloringwe show that any simple _n_-uniform non-_r_-colorable hypergraph _H_ has maximum edge degree at least _Δ(H) ≥ с · nr_<sup>_n−1_</sup> where _c_ > 0 is an absolute constant. We also give some applications of our probabilistic technique, we establish a new lower bound for the Van der Waerden number and extend the main result to the _b_-simple case. The work of the second author was supported by Russian Foundation of Fundamental Research (grant № 12-01-00683-a), by the program “Leading Scientific Schools” (grant no. NSh-2964.2014.1) and by the grant of the President of Russian Federation MK-692.2014.1

1. Dmitry Shabanov
joint work with
Jakub Kozik
Workshop on Extremal Graph Theory,
June 06, Yandex

2. Definitions
• A hypergraph 𝐻 = (𝑉, 𝐸) is a vertex set 𝑉 and a family of
subsets 𝐸 ⊂ 2 𝑉 whose elements are called the edges of the
hypergraph.
• A hypergraph 𝐻 = 𝑉, 𝐸 is said to be 𝑛-uniform if every edge
consists of exactly 𝑛 vertices.
• Let 𝐻 = (𝑉, 𝐸) be a hypergraph. A vertex coloring is called
proper for 𝐻 if there is no monochromatic edges in this
coloring.
• A hypergraph is said to be 𝑟-colorable if there is a proper
coloring with 𝑟 colors for it.
• The chromatic number of the hypergraph 𝐻, denoted by
𝜒(𝐻), is the minimum number of colors required for a proper
coloring.

3. Colorings of Graphs
The bound is tight (e.g., complete graphs or odd cycles).
This bound is tight up to the constant factor.
Basic fact
Any graph 𝐺 with maximum vertex degree Δ has chromatic
number at most Δ + 1.
Theorem (A. Johansson, 1996)
Any triangle-free graph 𝐺 with maximum vertex degree Δ has
chromatic number at most O(Δ/ln Δ).

4. What about hypergraphs?
• Let 𝐻 = (𝑉, 𝐸) be a hypergraph. The degree of an edge 𝐴 ∈ 𝐸
is the number of other edges of 𝐻 intersecting 𝐴. The
maximum edge degree of 𝐻 is denoted by Δ(𝐻).
This theorem was historically the first application of the famous
Local Lemma.
Theorem (P. Erdős, L. Lovász, 1973)
If 𝐻 is a 𝑛-uniform hypergraph with maximum edge degree
Δ(𝐻) at most
Δ 𝐻 ≤
1
𝑒
𝑟 𝑛−1
then 𝐻 is 𝑟-colorable (i.e. 𝜒 𝐻 = 𝑂( Δ 𝐻
1
𝑛−1
).

5. Theorem for 2-colorings
The proof is based on the random recoloring method.
Theorem (J. Radhakrishnan, A. Srinivasan, 2000)
If 𝐻 is an 𝑛-uniform hypergraph with maximum edge degree
Δ(𝐻) at most
Δ 𝐻 ≤ 0.17
𝑛
ln 𝑛
1
2
2 𝑛−1
then 𝐻 is 2-colorable.

6. Recent progress
The proof is based on Pluhár’s criterion for 𝑟-colorability of an
arbitrary hypergraph in terms of ordered 𝑟-chains.
UPDATE: Theorem holds for arbitrary number of colors 𝑟 with
𝑐 𝑟 replaced by an absolute constant 𝑐.
Theorem (D. Cherkashin, J. Kozik, 2013)
Suppose 𝑟 is fixed. Then for any sufficiently large 𝑛, if 𝐻 is an
𝑛-uniform hypergraph with Δ(𝐻) at most
Δ 𝐻 ≤ 𝑐 𝑟
𝑛
ln 𝑛
𝑟−1
𝑟
𝑟 𝑛−1
then 𝐻 is 𝑟-colorable.

7. Colorings of simple hypergraphs
• A hypergraph 𝐻 = (𝑉, 𝐸) is called simple if every two of its
distinct edges have at most one common vertex (as in graphs),
i.e. for any 𝐴, 𝐵 ∈ 𝐸, 𝐴 ≠ 𝐵, |𝐴 ∩ 𝐵| ≤ 1.
It turns out that it is somehow easier to color simple
hypergraphs.
Theorem (Z. Szabó, 1990)
For any 𝜀 > 0, there is 𝑛0 such that for any 𝑛 > 𝑛0, the
following statement holds: if 𝐻 is an 𝑛-uniform simple
hypergraph with Δ(𝐻) at most
Δ 𝐻 ≤ 2 𝑛−1
𝑛1−𝜀
then 𝐻 is 2-colorable.

8. Recent improvements
Theorem (A. Kupavskii, D. Shabanov, 2013)
For any 𝑛, 𝑟 ≥ 2, if 𝐻 is an 𝑛-uniform simple hypergraph with
Δ 𝐻 ≤ 𝑐
𝑛 ln ln 𝑛 2
ln 𝑛
𝑟 𝑛−1,
then 𝐻 is 𝑟-colorable.
Theorem (J. Kozik, 2013)
If 𝑛 > 𝑛0(𝑟) and 𝐻 is an 𝑛-uniform simple hypergraph with
Δ 𝐻 ≤ 𝑐
𝑛
ln 𝑛
𝑟 𝑛−1,
then 𝐻 is 𝑟-colorable.

9. Main new result
The proofs of the mentioned theorems appeared to be
“orthogonal”. The joined efforts led to the following result.
Theorem holds for any 𝑛 and 𝑟 as in the classical Theorem of
Erdős and Lovász.
Theorem (J. Kozik, D. Shabanov, 2014+)
If 𝐻 is an 𝑛-uniform simple hypergraph with maximum edge
degree Δ(𝐻) at most
Δ 𝐻 ≤ 𝑐 ⋅ 𝑛 𝑟 𝑛−1,
where 𝑐 > 0 is an absolute constant, then 𝐻 is 𝑟-colorable.

10. Comparison with other results
• We proved that any 𝑛 -uniform simple non- 𝑟 -colorable
hypergraph 𝐻 satisfies
Δ 𝐻 ≥ 𝑐 ⋅ 𝑛 𝑟 𝑛−1.
• If 𝑟 is a constant then this lower bound is 𝑛 times smaller
than the upper bound given by A. Kostochka and V. Rödl who
showed that there exists an 𝑛-uniform simple non-𝑟-colorable
hypergraph 𝐻 with
Δ 𝐻 ≤ 𝑛2
𝑟 𝑛−1
ln 𝑟 .
• For large 𝑟, A. Frieze and D. Mubayi established that any 𝑛-
uniform simple non-𝑟-colorable hypergraph 𝐻 satisfies
Δ 𝐻 ≥ 𝑐 𝑛 𝑟 𝑛−1
ln 𝑟
with 𝑐 𝑛 = 𝑂(𝑛2−2𝑛
).

11. Property B conjecture
Let 𝑚(𝑛) denote the minimum possible number of edges in an
𝑛-uniform non-2-colorable hypergraph.
Similar problem: Let Δ(𝑛) denote the minimum possible value
of the maximum edge degree in an 𝑛-uniform non-2-colorable
hypergraph.
Conjecture: Δ 𝑛 = Θ 𝑛2 𝑛 .
We proved the lower bound in the class of simple hypergraphs.
Conjecture (P. Erdős, L. Lovász, 1973)
𝑚 𝑛 = Θ 𝑛2 𝑛 .

12. Ingredients of the proof
• Random recoloring method
• Almost complete analysis of the recoloring
procedure (h-tree construction)
• Special variant of the Local Lemma

13. Random recoloring method
Suppose 𝐻 = (𝑉, 𝐸) is an 𝑛-uniform simple hypergraph. Without
loss of generality assume that 𝑉 = {1, … , 𝑚}.
Let 𝑋1, … , 𝑋 𝑚 be independent random variables with uniform
distribution on [0,1], weights of the vertices. Let 𝑝 ∈ (0,1) be a
real number.
A vertex 𝑣 ∈ 𝑉 is called a free vertex if 𝑋 𝑣 ≤ 𝑝. Only free vertices
are allowed to recolor during the recoloring procedure.
FIRST STAGE
Color every vertex randomly and independently with 𝑟 colors.
The obtained coloring is called initial.

14. Recoloring procedure
SECOND STAGE
1. Start with initial coloring.
2. If in the current coloring there is a monochromatic edge 𝐴 (of
some color 𝛼) containing a free vertex which has not been
recolored yet, then
– take a free vertex 𝑣 of 𝐴 with initial color 𝛼 and the least
weight 𝑋 𝑣;
– recolor 𝑣 with color 𝛼 𝑚𝑜𝑑 𝑟 + 1.
3. Repeat step 2 until there is no monochromatic edges with
non-recolored free vertices.

15. Recoloring procedure
In such situation we say that the third vertex blames the
edge 𝐴.

16. Construction of an h-tree
Suppose that recoloring procedure fails and in the final coloring
there is a monochromatic edge 𝐴 (root of the directed tree) of
some color 𝛼. Then every vertex of 𝐴
• either has initial color 𝛼 and is not free
• or has initial color 𝛼 − 1, is free and was recolored with 𝛼
during the recoloring process
Every vertex of the second type 𝐴
blames some other edge
(choose one for every vertex).
Let 𝐵1, … , 𝐵𝑠 be these edges.
We add them to the h-tree
as neighbors of 𝐴. 𝐵1, … , 𝐵𝑠

17. Construction of an h-tree
Every edge from 𝐵1, … , 𝐵𝑠 became completely monochromatic of
a color 𝛼 − 1 at some step of the recoloring procedure. So, in
the initial coloring 𝐵𝑖 can contain the vertices of a color 𝛼 − 2
which were recolored with 𝛼 − 1. Every such vertex blames
some edges (choose one for every vertex), we add them as
neighbors of 𝐵𝑖 in the h-tree. Continue the process if possible.
OBSERVATIONS
1. The edges 𝐶𝑖,1, … , 𝐶𝑖,𝑗 are different 𝐵𝑖
for every 𝑖.
2. The edges 𝐶𝑖,𝑟 and 𝐶 𝑘,𝑑 can
coincide for different 𝑖 ≠ 𝑘.
3. The leaves of the h-tree are
monochromatic in the initial coloring. 𝐶𝑖,1, … , 𝐶𝑖,𝑗

18. Analysis of bad events
There could be the following configurations in the h-tree.
1. The edges of the h-tree form a real hypertree.
2. There are cycles in the h-tree. In this case
we take the smallest subtree containing a cycle.
𝐴
Then either there is a short cycle (of length
≤ 2 ln 𝑛) or there is a large acyclic subtree
(of length > ln 𝑛).

19. Local Lemma
Theorem (Local Lemma, polynomial style)
Suppose that 𝑋1, … , 𝑋 𝑁 are independent random variables
and 𝐴1, … , 𝐴 𝑀 are events from the algebra generated by
them. Let 𝑣(𝐴𝑖) denote the smallest set of variables
𝑋𝑗 such that 𝐴𝑖 ∈ 𝜎(𝑋𝑗, 𝑗 ∈ 𝑣(𝐴𝑖)). Denote for 𝑗 = 1, … , 𝑁,
𝑤𝑗 𝑧 =
𝐴:𝑋 𝑗∈ 𝑣 𝐴
Pr 𝐴 𝑧 𝑣(𝐴)
.
Suppose that there exists a polynomial 𝑤 𝑧 such that
𝑤 𝑧 ≥ 𝑤𝑗 𝑧 for every 𝑗 and 𝑧 ≥ 1. If, moreover, there is
a real number 𝜏 ∈ (0,1) such that 𝑤
1
1−𝜏
≤ 𝜏, then
Pr 𝑖=1
𝑀
𝐴𝑖 > 0.

20. Application: Van der Waerden
Number
The function 𝑊(𝑛, 𝑟) from the Van der Waerden Theorem is
called the Van der Waerden function or the Van der Waerden
number.
Question: how can we estimate 𝑊 𝑛, 𝑟 ?
Theorem (B. Van der Waerden, 1927)
For any integers 𝑛 ≥ 3, 𝑟 ≥ 2, there exists the smallest
number 𝑊(𝑛, 𝑟) such that in any 𝑟-coloring of the set of
integers {1, … , 𝑊(𝑛, 𝑟)} there is a monochromatic arithmetic
progression of length 𝑛.

21. Known bounds for W(n,r)
The best general upper bound was obtained by W.T. Gowers
(2001):
𝑊 𝑛, 𝑟 ≤ 22 𝑟22 𝑛+9
.
In the particular cases 𝑛 = 3,4 the best results are due to T.
Sanders (2011), B. Green and T. Tao (2009).
Known lower bounds are very far away from Gowers’ tower.
E. Berlekamp (1968) 𝑊 𝑝 + 1,2 ≥ 𝑝2 𝑝
, 𝑝 is a prime.
Z. Szabó (1990) 𝑊 𝑛, 2 ≥ 2 𝑛 𝑛−𝜀, provided 𝑛 > 𝑛0(𝜀)

22. New lower bound for W(n,r)
This improves the previous results for 𝑊 𝑛, 2 of the type
2 𝑛 𝑛−𝜀.
When the number of colors is large in comparison with
progression length (say, 𝑟 > 2 𝑐 𝑛 ln𝑛 ), a better lower bound can
be obtained by using Hypergraph Symmetry Theorem.
Theorem (J. Kozik, D. Shabanov, 2014+)
For any 𝑛 ≥ 3, 𝑟 ≥ 2,
𝑊 𝑛, 𝑟 ≥ 𝑐 ⋅ 𝑟 𝑛−1,
where 𝑐 > 0 is an absolute constant.

23. Ideas of the proof
• We have to show that the hypergraph of arithmetic
progressions (vertex set = {1, … , 𝑁}, edges are arithmetic
progressions of length 𝑛) is 𝑟-colorable.
• This hypergraph of arithmetic progressions is not simple, but
(in some sense) close to be simple.
• Its codegree is at most 𝑛2
(not 1, as in simple hypergraphs)
which is sufficient for our probabilistic construction.
• Use the same random recoloring procedure.
• We have to deal with 2-cycles in
h-trees, especially with situations
when there are two edges with
a lot of common vertices (> 𝑛/2).