651 research outputs found
The true complexity of a system of linear equations
It is well-known that if a subset A of a finite Abelian group G satisfies a
quasirandomness property called uniformity of degree k, then it contains
roughly the expected number of arithmetic progressions of length k, that is,
the number of progressions one would expect in a random subset of G of the same
density as A. One is naturally led to ask which degree of uniformity is
required of A in order to control the number of solutions to a general system
of linear equations. Using so-called "quadratic Fourier analysis", we show that
certain linear systems that were previously thought to require quadratic
uniformity are in fact governed by linear uniformity. More generally, we
conjecture a necessary and sufficient condition on a linear system L which
guarantees that any subset A of F_p^n which is uniform of degree k contains the
expected number of solutions to L.Comment: 30 page
What is good mathematics?
Some personal thoughts and opinions on what ``good quality mathematics'' is,
and whether one should try to define this term rigorously. As a case study, the
story of Szemer\'edi's theorem is presented.Comment: 12 pages, no figures. To appear, Bull. Amer. Math. So
Packing tight Hamilton cycles in 3-uniform hypergraphs
Let H be a 3-uniform hypergraph with N vertices. A tight Hamilton cycle C
\subset H is a collection of N edges for which there is an ordering of the
vertices v_1, ..., v_N such that every triple of consecutive vertices {v_i,
v_{i+1}, v_{i+2}} is an edge of C (indices are considered modulo N). We develop
new techniques which enable us to prove that under certain natural
pseudo-random conditions, almost all edges of H can be covered by edge-disjoint
tight Hamilton cycles, for N divisible by 4. Consequently, we derive the
corollary that random 3-uniform hypergraphs can be almost completely packed
with tight Hamilton cycles w.h.p., for N divisible by 4 and P not too small.
Along the way, we develop a similar result for packing Hamilton cycles in
pseudo-random digraphs with even numbers of vertices.Comment: 31 pages, 1 figur
Ramsey numbers of Berge-hypergraphs and related structures
For a graph , a hypergraph is called a Berge-,
denoted by , if there exists a bijection such
that for every , . Let the Ramsey number
be the smallest integer such that for any -edge-coloring of
a complete -uniform hypergraph on vertices, there is a monochromatic
Berge- subhypergraph. In this paper, we show that the 2-color Ramsey number
of Berge cliques is linear. In particular, we show that for and where is a Berge-
hypergraph. For higher uniformity, we show that for
and for and sufficiently large. We
also investigate the Ramsey number of trace hypergraphs, suspension hypergraphs
and expansion hypergraphs.Comment: Updated to include suggestions of the refere
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