see reviews/interviews.

"It is shown that at least 50% of the probability mass of a sum of independent Rademacher random variables is within one standard deviation from its mean. This lower bound is sharp, it is much better than for instance the bound that can be obtained from application of the Chebishev inequality..."

"An alternate form for the binomial tail is presented, which leads to a variety of bounds for the central tail. A few can be weakened into the corresponding Chernoff and Slud bounds, which not only demonstrates the quality of the presented bounds, but also provides alternate proofs for the classical bounds."

"The {em longest common subsequence (LCS)} problem is a classic and well-studied problem in computer science. Palindrome is a word which reads the same forward as it does backward. The {em longest common palindromic subsequence (LCPS)} problem is an interesting variant of the classic LCS problem which finds the longest common subsequence between two given strings such that the computed subsequence is also a palindrome. In this paper, we study the LCPS problem and give efficient algorithms to solve this problem. To the best of our knowledge, this is the first attempt to study and solve this interesting problem."

"The k-domination number of a graph is the minimum size of a set X such that every vertex of G is in distance at most k from X. We give a linear time constant-factor approximation algorithm for k-domination number in classes of graphs with bounded expansion, which include e.g. proper minor-closed graph classes, classes closed on topological minors or classes of graphs that can be drawn on a fixed surface with bounded number of crossings on each edge.

The algorithm is based on the following approximate min-max characterization. A subset A of vertices of a graph G is d-independent if the distance between each pair of vertices in A is greater than d. Note that the size of the largest 2k-independent set is a lower bound for the k-domination number. We show that every graph from a fixed class with bounded expansion contains a 2k-independent set A and a k-dominating set D such that |D|=O(|A|), and these sets can be found in linear time. For domination number (k=1) the assumptions can be relaxed, and the result holds for all graph classes with arrangeability bounded by a constant."

used to indicate possession
(actually a hiragana)

This post follows a recent paper On sunflowers  and matrix multiplication by Noga Alon, Amir Spilka, and Christopher Umens (ASU11) which rely on an earlier paper Group-theoretic algorithms for matrix multiplication, by Henry Cohn, Robert Kleinberg, Balasz Szegedy, and Christopher Umans (CKSU05), and refers also to a paper by Don Coppersmith and Shmuel Winograd (CW90). 

Three famous problems
The Erdos-Rado sunflower conjecture
The Erdos-Rado sunflower (Delta system) theorem and conjecture was already menioned in this post on extremal set theory.

A sunflower (a.k.a. Delta-system) of size is a family of sets such that every element that belongs to more than oneofthe sets belongs to all of them. A basic and simple result of Erdos and Rado asserts that

Erdos-Rado sunflower theorem: There is a function so that every family of -sets with more than  members contains a sunflower of size .

One of the most famous open problems in extremal combinatorics is:

The Erdos-Rado conjecture: Prove that .

Here, is a constant depending on . This is most interesting already for .

Three term arithmetic progressions
The cup set problem (three terms arithmetic progressions in ):

The cup set problem was also discussed here quite extensively. (See, e.g. this post.)

Let . The cap set problem  asks for the maximum number of elements in a subset of  which contains no arithmetic progression of size three or, alternatively, no three vectors that sum up to 0(modulo 3). (Such a set is called a cup set.) If is a cap set of maximum size we can ask how the function behaves. Roy Meshulam proved, using Roth’s argument, that . Edell found an example of a cap set of size . So .  The gap is exponential.  

The strong cap set conjecture: for some .

Of course, the cap set problem is closely related to

Erdos-Turan problem (for ): What is the larget size  of a subest of {1,2,…,n} without 3-term arithmetic progression?

Matrix multiplications
Let ω be the smallest real number so that there is an algorithm for multiplying  two matrices which requires arithmetic operations.  

The ω=2 conjecture: ω=2.

A very recent breakthrough by Virginia Vassilevska Williams (independently) following an earlier breakthrough by Andrew Stothers improved the Coppersmith-Winograd algorithm which gave ω =2.376, to 2.374 and 2.373 respectively. (See the discussions over Lipton’s blog (1,2), Shtetl optimized, and Computational Complexity.)

It turns out that these three conjectures are related. (The connection of matrix multiplication and the Erdos-Turan problem is fairly old, but I am not sure what an even drastic improvment of Behrends’s lower bound would say about .)

Three combinatorial conjectures that imply ω=2.
Remarkably, an affarmative answer for the ω=2 conjecture would folow from each one of three combinatorial conjectures. One conjecture goes back to CW90 and two were described in CKSU05. I will not present the precise formulations in order to encourage the readers to look at the original papers. (Maybe I will add the formulations later.)

The no disjoint equivoluminous subsets conjecture (CW90).

The Strong UPS conjecture (CKSU05).

Theorem: Conjecture CW90 implies the strong UPS conjecture.

CKSU’s two-family conjecture (CKSU05).

Relations between these problems
Here are some results taken from ASU11 about the relations between these combinatorial questions. The first result goes back to Erdos and Szemeredi.

The weak sunflower conjecture: A family of subsets of {1,2,…,n}  with no sunflower of size 3 can have at most sets.

The following results are not difficult.

Theorem: The strong sunflower conjecture implies the weak sunflower conjecture.

Theorem: The strong cup set conjecture also implies the weak sunflower conjecture.

Theorem: The weak sunflower conjecture implies that the CW90 conjecture is false. 

It follows that CW90 conjecture is in conflict both with the Erdos Rado sunflower conjecture and with the strong cup set conjecture.

Theorem: The strong cup set conjecture implies that the strong UPS conjecture is false.

While two family theorems are quite popular in extremal combinatorics (see this post and this one), CKSU’s two family conjecture is still rather isolated from other combinatorics.

What to believe?
This is a nice topic for discussion.

A very slick way to run tons of different regressions, mining your way toward the best model

"Every time you shuffle a deck of playing cards, it's likely that you have come up with an ordering of cards that is unique in human history."

"We study the reverse mathematics of the theory of countable second-countable topological spaces, with a focus on compactness. We show that the general theory of such spaces works as expected in the subsystem $mathsf{ACA}_0$ of second-order arithmetic, but we find that many unexpected pathologies can occur in weaker subsystems. In particular, we show that $mathsf{RCA}_0$ does not prove that every compact discrete countable second-countable space is finite and that $mathsf{RCA}_0$ does not prove that the product of two compact countable second-countable spaces is compact. To circumvent these pathologies, we introduce strengthened forms of compactness, discreteness, and Hausdorffness which are better behaved in subsystems of second-order arithmetic weaker than $mathsf{ACA}_0$."