POLYMATH!
Noga Alon Birthday Conference
Gil Kalai
Einstein Institute of Mathematics
Hebrew University of Jerusalem.
Noga’s formulas
1) Seperators for Minor-free graphs
(Bhargav Narayanan)
2) Aaaalmost Stanley-Wilf conj. (Ian Alevy)
3) All pairs shortest path. (Josh Alman)
4) A bound on the linear arboricity la(G) of a
graph G of maximum degree d from a
paper published in 1988. (Ross Kang)
5) Maximum number of directed Hamiltonian
path in a tournament (Shakhar
Smorodinsky, Louis DeBiasio)
6) Shannon capacity of a union (Jean
Cardinal, Daniel Soltész)
7) (corrected) Turan number for very
asymmetric bipartite graphs (Adam
Sheffer x1.5) (Sebi Cioaba, Ferdinand
Ihringer)
8) Largest number of edges in a bipartite
subgraph (Sebi Cioaba)
Noga’s formulas 2
1)
2)
Alon-Boppana (Lior Silberman) x1.5
Ramsey number R(C4, C4, C4, Kn )
(Sushant Sachdeva &David Conlon)
3) The discrete Cheeger-Buser inequality.
(Emmanuel Kowalski x2)
4) Monotone circuits for cliques (Gautam
Kamath)
5) Maximum number of copies of a graph H in a
graph with l edges. (Yufei Zhao x2)
6) 123-theorem of Alon-Yuster (Nicolas Rivera)
7) Lower bound an restricted sum of A and B.
(Shachar Lovett)
Noga and I: academic twins
Ph. D. Students of
Micha A. Perles
Altshuler (**)
Katchalski (74)
Koren (7?)
Kallay (79)
Linial (80)
Shemer (82)
Alon (83)
Kalai (83)
Kupitz (85)
Adin(*) (93)
Smilanski
Deutch
Pinchasi (*)
Magazanik (07)
Sigron
Nitzan (*)
Kammal
Keller
Polymath
A polymath (Greek: πολυμαθής, polymathēs,
having learned much") is a person whose expertise
spans a significant number of different subject
areas; such a person is known to draw on complex
bodies of knowledge to solve specific problem
the idea of the polymath was expressed by Leon
Battista Alberti (1404–1472), in the statement,
most suitable to Noga
"a man (who) can do all things if
he will".
3-regular subgraphs
Conjecture (Berge): Every simple 4-regular graph
contains a 3-regular subgraph.
Theorem (Alon, Friedland and Kalai, 1984): Every 4regular graph plus an edge contains a 3-regular graph.
Theorem (AFK, 84) A graph with n vertices and 2n+1
edges contains a nonempty subgraph with all degrees
0(mod 3).
Collaboration
• Gil’s top coauthors
1) Kahn 8
2) Alon 7
3) Meshulam 5
4-6) Bourgain, Friedgut, Linial 4
7-9) Bjorner, Keller, Mossel 3
10-24) Bar-Hillel, Bar-Nathan,
Benjamini (*), Friedland,
Kleinschmidt, Matouskek (*),
McKay, Meisinger, Nisan, Nevo(*),
Novik(*), Richlin, Schramm(*),
Stockmayer, Tennenholtz(*) 2
• Noga’s top coauthors (incomplete)
1) Krivelevich 32 (AKS 9)
2) Sudakov 29
3)
Yuster 20
4) Azar 17
5) Tennenholz 14
6-7) Kleitman, Shapira 13
7-9) Lubetzky, Rodl , Spencer 11
10)
Caro 9
11-12) Feldman, Seymour 8
13-16) Fischer, Furedi , Kostochka, Stav 7
17-23) Asodi, Bollobas, Erdos, Kalai, Linial,
M. Naor , M. Szegedi 6
The polynomial method
3-regular subgraphs (2)
Taskinov proved Berge’s conjecture using
direct graph theoretical arguments.
Rodl and Szemeredi proved that a graph
with n vertices and Cnlogn edges always
contains a 3-regular graph, but
constructed examples of graphs with
C’nloglogn edges without a 3-regular
subgraph.
Regular subgraphs: problems
Does every 4-regular graph plus an edge (or 5 regular graph, or just
4-regular graph) have a type I (3-edge colorable) 3-regular
subgraph?
Is there a polynomial-time algorithm to find a subgraph with all
degrees divisible by 3 in a graph with n vertices and 2n+1 edges.
How many edges guarantee a 3-regular subgraph?
What about subgraphs with vertex degrees divisible by m, when m
is not a prime-power? How many vectors in (Z/mZ)n guarantees a
subset summing up to 0?
Shellability and upper bound
theorems
Some of the places where our
roads crossed again
Few polytopes; many spheres: Goodman Pollack (85),
Alon(87), Kalai (88)
Transversal theorems: (Alon & Kalai (95) , Alon, Kalai,
Matousek, Mashulam (02)) (Following AlonKleitman’s solution of Hadwiger-Debrunner’s
conjecture.)
Spectral methods, Collective coin flipping, mobile user
tracking and distributed job scheduling, social choice,
hereditary discrepancy of HAP.
JERUSALEM COMBINATORICS 1993
Polymath (modern use)
Polymath projects are a form of open Internet collaboration
aimed towards a major mathematical goal, usually to settle a
major mathematical problem.
This is a concept introduced in 2009 by Tim Gowers who
described it as “a large collaboration in which no single person
has to work all that hard.”
It is in line with other forms of open Internet research activity
which include MathOverflow and are sometimes referred to
as Science 2.0.
“Sometimes it is not market forces that achieve efficiencies,
but cooperatives.” (Nature, editorial 2014)
Polymath projects
(*) Polymath1 (Density Hales Jewett); (TG)
Polymath3 (Polynomial Hirsch Conjecture) (GK)
Polymath4 (Deterministically finding primes)(TT)
(*) Polymath 5 (Erdos discrepancy problem) (TG)
Polymath 7 (Hot spots) (TT)
(*) Polymath 8a and 8b (Gaps in primes) (TT)
Polymath 10 (Sunflower conjecture) (GK)
A few dozens polymath proposals.
The IAS polymath debate (2010)
The polymath format could be a
better way of doing mathematics
“If the polymath model were to grow to dominate
mathematics, younger mathematicians would be
driven in the direction of online collaboration at the
cost of traditional mathematics, which has produced
the likes of Alexander Grothendieck, whose individual
contributions to the field have been revolutionary.”
“There is no danger and
no hope that online
collaboration will
supersede lone
endeavors” (but I like it)
Interesting, but runs
the danger of making
us “managers” rather
than researchers
Polymath10: The Erdos-Rado
Sunflower Conjecture
A collection of sets is a sunflower if every
element which is included in two sets is
included in all of them.
The sets are called petals; their intersection is
called the head.
Early heroes: Abott, Hanson and Sauer,
Spencer; Kostochka
The conjecture
Let f(k,r) denotes the maximum size of a family
of k-sets without a subflower of size r.
Erdős-Rado's sunflower conjecture:
There is a constant C=Cr such that
f(k,r) < Crk.
Balanced families
A family of k-sets is balanced if we can color the
ground set with k colors such that every set is
colorful.
Let fb (k,r) denotes the maximum size of a family
of k-sets without a sunflower of size r.
Lemma:
f(k,r) < ek fb (k,r).
Problem: exact lower bounds for balanced subhypergraphs of
hypergraphs with e edges like Noga’s formula 8.
A refinement
Let f(k,r,n,m) be the maximum number of ksubsets from [n] without a sunflower with r
petals whose head has less than m elements.
For r=2 and arbitrary m, and for m=1 and
arbitrary r we get well-known questions of
Erdos-Ko-Rado type.
Homology
Let F be a family of sets, let K be the simplicial
complex spanned by F and let Hi (K) be the ith
(reduced) homology group of K.
Homology
chains
Weighted homology
Cycles for multiple boundaries
Homology for multiple boundaries
Conjectures
Some partial results
Are available.
Analogous results are available (when you replace
[2,1] by [1,1] and [2k,k] by [k,k]) , but
The conjecture is largely a wishful thinking, so far.
More from polymath10
• Phillip Gibbs found a better connection between the balanced and general
cases. If the sunflower conjecture is true then
f(k,r) < (1+o(1))k fb (k,r).
• Dömötör Pálvölgyi shot down various working conjectures, and raised
interesting questions.
• Some computer experimentation was made by several participants.
• Tim Gowers proposed a “pseudorandom” approach
• Shachar Lovett commented on a special case related to matrix
multiplication
• Ferdinand Ihringer found inequalities for the refined version.
• Eran Nevo showed that for balanced families, intersecting implies acyclic.
Extensions of this result are crucial for partial results towards Conjecture A .
• Karim Adiprasito discussed the Cohen-Macaulay case
• Some new extremal results for balanced families were found.
Conjectures (late update)
Some partial results
Are available.
Analogous results are available, but
The conjecture is largely a wishful
thinking, so far.
Are polymath projects, and, more
generally, “Science 2.0” viable ideas?
1. Putting aside questions about incentives, can
it lead to a better way of doing science (at
least in some cases)? or to other major fruits?
2. What about incentives?
Social choice
Both Noga and me did some work on social
choice: question about aggregating
information and preferences of many
individuals.
Large committees require large budget!
Given a committee of k members (k an odd integer)
that need to give m followships. (The number of
applications is unlimited). Is there a rule that
exclude the possibility of a candidate not getting the
fellowship in spite having the majority of the
committee preferring her to all those who received
the fellowship.
• Condorcet: No: for k=3, m=1.
• Alon, Brightwell, Kierstead, Kostochka and
Winkler: NO for m< Ck/ log k;
YES for m>C k log k.
Question: what about general voting rules instead of majority?
Condorcet
1743-1794
Happy birthday dear Noga