Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0707.4269

Structure and randomness in combinatorics

Terence Tao
View original source ↗Read source document ↗Export metadata ↓

Abstract

Combinatorics, like computer science, often has to deal with large objects of unspecified (or unusable) structure. One powerful way to deal with such an arbitrary object is to decompose it into more usable components. In particular, it has proven profitable to decompose such objects into a \emph{structured} component, a \emph{pseudo-random} component, and a \emph{small} component (i.e. an error term); in many cases it is the structured component which then dominates. We illustrate this philosophy in a number of model cases.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Terence Tao. 2007-08-03. Structure and randomness in combinatorics. https://arxiv.org/abs/0707.4269

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Existence of 3 anti-cocircular truncated Möbius planes and constructions of strength-4 covering arrays

Two projective (affine) planes with the same point sets are orthogoval if the common intersection of any two lines, one from each, has size at most two. The existence of a pair of orthogoval projective planes has been proven and published independently many times. A strength-$t$ covering array, denoted by CA$(N; t, k, v)$, is an $N \times k$ array over a $v$-set such that in any $t$-set of columns, each $t$-tuple occurs at least once in a row. A pair of orthogoval projective planes can be used to construct a strength-$3$ covering array CA$(2q^3-1; 3, q^2 + q + 1, q)$. Our work extends this result to construct arrays of strength $4$. A $k$-cap in a projective geometry is a set of $k$ points no three of which are collinear. In $PG(3,q)$, an ovoid is a maximum-sized $k$-cap with $k =q^2+1$. Its plane sections (circles) are the blocks of a $3-(q^2 + 1, q + 1, 1)$ design, called a Möbius plane of order $q$. For $q$ an odd prime power, we prove the existence of three truncated Möbius planes, such that for any choice of these circles, one from each plane, their intersection size is at most three. From this, we construct a strength-$4$ covering array CA$(3q^4-2; 4, \frac{q^2+1}{2}, q)$. For $q \geq 11$, these covering arrays improve the size of the best-known covering arrays with the same parameters by almost 25 percent. The CA$(3q^4 -3; 4, \frac{q^2 +1}{2}, q)$ is used as the main ingredient in a recursive construction to obtain a CA$(5q^4 - 4q^3 - q^2 + 2q; 4, q^2 +1, q)$. Some improvements are obtained in the size of the best-known arrays using these covering arrays.

math.CO↗
arXiv2026

Modular Constructions of g-Golomb Rulers

A set \(\mathcal{G}\) of integers is a \(g\)-Golomb ruler if each positive difference appears at most \(g\) times between any 2 elements of the set, and \(G(g,n)\) denotes the minimum diameter of such a ruler with \(n\) marks. We prove a general lemma for passing from certain modular constructions to ordinary \(g\)-Golomb rulers. The key point is that, in a modular \(g\)-Golomb ruler, no cyclic gap length can occur more than \(g\) times. This gives a larger guaranteed cut than the previous average gap argument. We apply this lemma to cyclic relative difference sets, Singer sets, Ruzsa--Spence rulers, and Paley quadratic residues to provide many competing constructions for \(g\)-Golomb Rulers. A computation on the grid \(1\le g\le500\), \(n=g+b\), \(2\le b\le500\), compares the four resulting construction families.

math.CO↗
arXiv2026

Optimal discrepancy for triangles

We prove that every $n$-point set in the plane has a two-coloring with discrepancy $O(n^{1/4})$ for all triangles. This improves the previous bound $O(n^{1/4}\sqrt{\log n})$ and matches the classical lower bound.

math.CO↗