Induced Subgraphs of Bounded Degree and Bounded Treewidth

We prove that for all integers $kgeq tgeq 0$ and $dgeq 2k$, every graph $G$ with treewidth at most $k$ has a `large\u27 induced subgraph $H$, where $H$ has treewidth at most $t$ and every vertex in $H$ has degree at most $d$ in $G$. The order of $H$ depends on $t$, $k$, $d$, and the order of $G$. With $t=k$, we obtain large sets of bounded degree vertices. With $t=0$, we obtain large independent sets of bounded degree. In both these cases, our bounds on the order of $H$ are tight. For bounded degree independent sets in trees, we characterise the extremal graphs. Finally, we prove that an interval graph with maximum clique size $k$ has a maximum independent set in which every vertex has degree at most $2k$

Szemer\u27edi\u27s regularity lemma revisited

Szemer\u27edi\u27s regularity lemma is a basic tool in graph theory, and also plays an important role in additive combinatorics, most notably in proving Szemer\\u27edi\u27s theorem on arithmetic progressions . In this note we revisit this lemma from the perspective of probability theory and information theory instead of graph theory, and observe a variant of this lemma which introduces a new parameter $F$. This stronger version of the regularity lemma was iterated in a recent paper of the author to reprove the analogous regularity lemma for hypergraphs

Uniquely circular colourable and uniquely fractional colourable graphs of large girth

Given any rational numbers $r \geq r\u27 >2$ and an integer $g$, we prove that there is a graph $G$ of girth at least $g$, which is uniquely $r$-colourable and uniquely $r\u27$-fractional colourable

On primitive symmetric association schemes with m_1=3

We classify primitive symmetric association schemes with m_1 = 3. Namely, it is shown that the tetrahedron, i.e., the association scheme of the complete graph K_4, is the unique such association scheme. Our proof of this result is based on the spherical embeddings of association schemes and elementary three dimensional Euclidean geometry

Best Simultaneous Diophantine Approximations under a Constraint on the Denominator

We investigate the problem of best simultaneous Diophantine approximation under a constraint on the denominator, as proposed by Jurkat. New lower estimates for optimal approximation constants are given in terms of critical determinants of suitable star bodies. Tools are results on simultaneous Diophantine approximation of rationals by rationals with smaller denominator. Finally, the approximation results are applied to the decomposition of integer vectors

Sum and product of different sets

Let A and B be two finite sets of numbers. The sum set and the product set of A, B are A + B = {a + b : a in A, b in B}, and AB = {ab : a in A, b in B}. $ We prove that A+B is as large as possible when AA is not too big. Similarly, AB is large when A+A is not too big. The methods rely on the Lambda_p constant of A, bound on the number of factorizations in a generalized progression containing A, and the subspace theorem