69 research outputs found
Equation-regular sets and the Fox–Kleitman conjecture
Given k ≥ 1, the Fox–Kleitman conjecture from 2006 states that there exists a nonzero
integer b such that the 2k-variable linear Diophantine equation
∑k
i=1
(xi − yi) = b
is (2k − 1)-regular. This is best possible, since Fox and Kleitman showed that for all
b ≥ 1, this equation is not 2k-regular. While the conjecture has recently been settled for
all k ≥ 2, here we focus on the case k = 3 and determine the degree of regularity of
the corresponding equation for all b ≥ 1. In particular, this independently confirms the
conjecture for k = 3. We also briefly discuss the case k = 4
Resolving a Conjecture on Degree of Regularity of Linear Homogeneous Equations
A linear equation is -regular, if, for every -coloring of the positive
integers, there exist positive integers of the same color which satisfy the
equation. In 2005, Fox and Radoicic conjectured that the equation , for any , has a degree of
regularity of , which would verify a conjecture of Rado from 1933. Rado's
conjecture has since been verified with a different family of equations. In
this paper, we show that Fox and Radoicic's family of equations indeed have a
degree of regularity of . We also provide a few extensions of this result.Comment: 8 page
On the degree of regularity of a certain quadratic Diophantine equation
We show that, for every positive integer r, there exists an integer b = b(r) such that the 4-variable quadratic
Diophantine equation (x1 − y1)(x2 − y2) = b is r-regular. Our proof uses Szemerédi’s theorem on arithmetic
progressions
The regularity method for graphs with few 4-cycles
We develop a sparse graph regularity method that applies to graphs with few 4-cycles, including new counting and removal lemmas for 5-cycles in such graphs. Some applications include:
* Every n-vertex graph with no 5-cycle can be made triangle-free by deleting o(n^(3/2)) edges.
* For r ≥ 3, every n-vertex r-graph with girth greater than 5 has o(n^(3/2)) edges.
* Every subset of [n] without a nontrivial solution to the equation x₁+x₂+2x₃ = x₄+3x₅ has size o(√n)
The regularity method for graphs with few 4-cycles
We develop a sparse graph regularity method that applies to graphs with few
4-cycles, including new counting and removal lemmas for 5-cycles in such
graphs. Some applications include:
* Every -vertex graph with no 5-cycle can be made triangle-free by
deleting edges.
* For , every -vertex -graph with girth greater than has
edges.
* Every subset of without a nontrivial solution to the equation has size .Comment: 23 page
A collection of open problems in celebration of Imre Leader's 60th birthday
One of the great pleasures of working with Imre Leader is to experience his
infectious delight on encountering a compelling combinatorial problem. This
collection of open problems in combinatorics has been put together by a subset
of his former PhD students and students-of-students for the occasion of his
60th birthday. All of the contributors have been influenced (directly or
indirectly) by Imre: his personality, enthusiasm and his approach to
mathematics. The problems included cover many of the areas of combinatorial
mathematics that Imre is most associated with: including extremal problems on
graphs, set systems and permutations, and Ramsey theory. This is a personal
selection of problems which we find intriguing and deserving of being better
known. It is not intended to be systematic, or to consist of the most
significant or difficult questions in any area. Rather, our main aim is to
celebrate Imre and his mathematics and to hope that these problems will make
him smile. We also hope this collection will be a useful resource for
researchers in combinatorics and will stimulate some enjoyable collaborations
and beautiful mathematics
Recommended from our members
Combinatorics and Probability
The main theme of this workshop was the use of probabilistic methods in combinatorics and theoretical computer science. Although these methods have been around for decades, they are being refined all the time: they are getting more and more sophisticated and powerful. Another theme was the study of random combinatorial structures, either for their own sake, or to tackle extremal questions. Both themes were richly represented at the workshop, with many recent exciting results presented by the lecturers
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