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Enumerative combinatorics, graph theory, order theory, posets, matroids, designs and other discrete structures. It also includes algebraic, analytic and probabilistic combinatorics.

14 votes
1 answer
797 views

Theorems proved using combinatorial nullstellensatz that have no other known proof

Alon's (or Alon and Tarsi's?) combinatorial nullstellensatz is a powerful algebraic tool with many applications in combinatorics and number theory. See this, this, this and this mathoverflow question. …
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11 votes

Important formulas in combinatorics

The Kneser graph $KG_{n,k}$ is the graph on $k$-subsets of $\{1, \dots, n\}$ with two subsets made adjacent when they are disjoint. The formula $$\chi(KG_{n,k}) = n - 2k + 2$$ was proved by Lovász in …
11 votes
2 answers
781 views

Blocking sets in three dimensional finite affine spaces

What is the smallest possible size of a set of points in $\mathbb{F}_q^3$ which intersects (blocks) every line? Clearly the union of three affine hyperplanes that intersect in a singleton, say $x = 0 …
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10 votes

How to recognise that the polynomial method might work

I would like to add some more examples and references for the so called polynomial method that can help us recognise when it can be applied. From what I understand so far, the polynomial method fall …
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9 votes
Accepted

The most number of points that realize only $k$ distinct distances

Bannai, Bannai and Stanton proved that $f_d(k) \leq {d + k \choose k}$ in 1983. See: http://link.springer.com/article/10.1007%2FBF02579288 I don't think this bound has been improved in general. It is …
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6 votes
1 answer
454 views

Applications of small Kakeya sets over finite fields

It was proved by Dvir that a Kakeya set in $\mathbb{F}_q^n$ has size at least $q^n/n!$, a bound which was later improved to $q^n/2^n$. For $n = 2$ and $q$ odd the exact bound is $q(q+1)/2 + (q-1)/2$ …
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5 votes
0 answers
232 views

A question on hyperplanes in partial linear spaces and hypergraphs

A partial linear space (or a linear hypergraph) is a point line geometry $(P,L,I)$ where for every pair of points there is at most one line incident with both of them. A hyperplane in a partial linear …
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5 votes
Accepted

$(n-2)$-blocking sets in $AG(n,2)$

Not much is known for the general case. Let $m(k, n, q)$ denote the minimum size of an $k$-blocking set in $AG(n, q)$. Trivially we have $m(0, n, q) = q^n$ and $m(n, n, q) = 1$. By Jamison/Brouwer-S …
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5 votes

Can Schwartz-Zippel be formulated for commutative rings instead of fields?

See Section 4 in "On Zeros of a Polynomial in a Finite Grid" to see how Schwartz-Zippel lemma and many similar results on zeros of polynomials work for arbitrary commutative rings as long as the "grid …
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5 votes

Combinatorial databases

Andries Brouwer's collection of strongly regular graphs: http://www.win.tue.nl/~aeb/graphs/srg/srgtab.html Eric Moorhouse's collections of finite projective planes and generalized polygons: http://eri …
4 votes

What are the applications of hypergraphs?

Every finite geometry (projective planes, generalized polygons, polar spaces, near polygons, etc.) and every block design (Witt design, difference sets, Steiner triple systems, etc.) is a hypergraph. …
4 votes
2 answers
380 views

Finding the set of all $0$-$1$ vectors in an affine subspace

We are given a $0$-$1$ matrix $A$ with constant row and column sum, and we need to find out if there exists a $0$-$1$ vector in the solution space of $Ax = \mathbf{1}$ over $\mathbb{Q}$ (or $\mathbb{Z …
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4 votes

Linear algebra proofs in combinatorics?

Here are some examples where the dimension of a vector space of polynomials is used to solve a combinatorial problem. Theorem 1 There are at most $n(n+1)/2$ equiangular lines in $\mathbb{R}^n$. Proof. …
3 votes

Blocking sets in three dimensional finite affine spaces

Here is an improvement of the upper bound which I found in ``The polynomial method in Galois geometries'' by Simeon Ball. See page number 4. The known constructions are somewhat crude. For exampl …
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3 votes

On MDS code property

It has been proved by Simeon Ball that for $k \leq p$, all $[n, k, n-k+1]_q$ codes are Reed-Solomon codes, where $q = p^h$. See Corollary 9.2 in the following paper: Ball, S. On sets of vectors of a …
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