Related papers: Partition theorems for Ketonen-Solovay largeness

Motivated by a partition inequality of Bessenrodt and Ono, we obtain analogous inequalities for $k$-colored partition functions $p_{-k}(n)$ for all $k\geq2$. This enables us to extend the $k$-colored partition function multiplicatively to a…

We prove several bounds on the number of incidences between two sets of multivariate polynomials of bounded degree over finite fields. From these results, we deduce bounds on incidences between points and multivariate polynomials, extending…

We address partition regularity problems for homogeneous quadratic equations. A consequence of our main results is that, under natural conditions on the coefficients $a,b,c$, for any finite coloring of the positive integers, there exists a…

A theorem of R\"odl states that for every fixed $F$ and $\varepsilon>0$ there is $\delta=\delta_F(\varepsilon)$ so that every induced $F$-free graph contains a vertex set of size $\delta n$ whose edge density is either at most $\varepsilon$…

We study the minimax problem for restricted two-fold self-sumsets in $k$-colorings of $\mathbb{Z}_n$. For primes $p$ with $2\le k\le p$ we determine the exact minimum $\max\{0,\,2\lceil p/k\rceil-3\}$. For general $n$ (with $m=\lceil…

Ramsey's theorem states that for all finite colorings of an infinite set, there exists an infinite homogeneous subset. What if we seek a homogeneous subset that is also order-equivalent to the original set? Let $S$ be a linearly ordered set…

Given a finite coloring (or finite partition) of the free semigroup $A^+$ over a set $A$, we consider various types of monochromatic factorizations of right sided infinite words $x\in A^\omega$. Some stronger versions of the usual notion of…

We propose an algebraic geometry framework for the Kakeya problem. We conjecture that for any polynomials $f,g\in\F_{q_0}[x,y]$ and any $\F_q/\F_{q_0}$, the image of the map $\F_q^3\to\F_q^3$ given by $(s,x,y)\mapsto…

A \emph{$k$--decomposition} of the complete graph $K_n$ is a decomposition of $K_n$ into $k$ spanning subgraphs $G_1,...,G_k$. For a graph parameter $p$, let $p(k;K_n)$ denote the maximum of $\displaystyle \sum_{j=1}^{k} p(G_j)$ over all…

In his recent work, Andrews revisited two-color partitions with certain restrictions on the differences between consecutive parts, and he established three theorems linking these two-color partitions with more familiar kinds of partitions.…

The famous partition theorem of Euler states that partitions of $n$ into distinct parts are equinumerous with partitions of $n$ into odd parts. Another famous partition theorem due to MacMahon states that the number of partitions of $n$…

Schmidt's theorem is significantly generalized, to partitions in which periodic but otherwise arbitrary subsets of parts are counted or uncounted. The identification of such sets of partitions with colored partitions satisfying certain…

Every infinite group $G$ of regular cardinality can be partitioned $G=A_1\cup A_2$ so that $G\neq FA_1$, $G\neq FA_2$ for every subset $F\subset G$ of cardinality $|F|<|G|$. The first author asked whether the same is true for each group $G$…

Motivated by a problem in additive Ramsey theory, we extend Todorcevic's partitions of three-dimensional combinatorial cubes to handle additional three-dimensional objects. As a corollary, we get that if the continuum hypothesis fails, then…

We show that if $P$ is a lattice polytope in the nonnegative orthant of $\R^k$ and $\chi$ is a coloring of the lattice points in the orthant such that the color $\chi(a+b)$ depends only on the colors $\chi(a)$ and $\chi(b)$, then the number…

There are a number of well-known problems and conjectures about partitioning graphs to satisfy local constraints. For example, the majority colouring conjecture of Kreutzer, Oum, Seymour, van der Zypen and Wood states that every directed…

The well known fractional Helly theorem and colorful Helly theorem can be merged into the so called colorful fractional Helly theorem. It states: For every $\alpha \in (0, 1]$ and every non-negative integer $d$, there is $\beta_{col} =…

In 1943, Hadwiger conjectured that every graph with no $K_t$ minor is $(t-1)$-colorable for every $t\ge 1$. In the 1980s, Kostochka and Thomason independently proved that every graph with no $K_t$ minor has average degree $O(t\sqrt{\log…

In a series of four papers we prove the following relaxation of the Loebl-Komlos-Sos Conjecture: For every $\alpha>0$ there exists a number $k_0$ such that for every $k>k_0$ every $n$-vertex graph $G$ with at least $(\frac12+\alpha)n$…

We analyze the Dual Ramsey Theorem for $k$ partitions and $\ell$ colors ($\mathsf{DRT}^k_\ell$) in the context of reverse math, effective analysis, and strong reductions. Over $\mathsf{RCA}_0$, the Dual Ramsey Theorem stated for Baire…