Related papers: On uncommon systems of equations
A system of linear equations $L$ is common over $\mathbb{F}_p$ if, as $n\to\infty$, any 2-coloring of $\mathbb{F}_p^n$ gives asymptotically at least as many monochromatic solutions to $L$ as a random 2-coloring. The notion of common linear…
A system of linear equations in $\mathbb{F}_p^n$ is \textit{common} if every two-colouring of $\mathbb{F}_p^n$ yields at least as many monochromatic solutions as a random two-colouring, asymptotically as $n \to \infty$. By analogy to the…
A linear equation with coefficients in $\mathbb{F}_q$ is common if the number of monochromatic solutions in any two-coloring of $\mathbb{F}_q^n$ is asymptotically (as $n \to \infty$) at least the number expected in a random two-coloring.…
A linear configuration is said to be common in $G$ if every 2-coloring of $G$ yields at least the number of monochromatic instances of a randomly chosen coloring. Saad and Wolf asked whether, analogously to a result by Thomason in graph…
A linear configuration is said to be common in a finite Abelian group $G$ if for every 2-coloring of $G$ the number of monochromatic instances of the configuration is at least as large as for a randomly chosen coloring. Saad and Wolf…
A system of linear equations in $\mathbb{F}_p^n$ is \textit{Sidorenko} if any subset of $\mathbb{F}_p^n$ contains at least as many solutions to the system as a random set of the same density, asymptotically as $n\to \infty$. A system of…
We study the number of monochromatic solution to linear equation in $\{1,\dots,n\}$ when we color the set by at least three colors. We consider the $r$-commonness for $r\geq 3$ of linear equation with odd number of terms, and we also prove…
We study the number of monochromatic solutions to linear equations in a $2$-coloring of $\{1,\ldots,n\}$. We show that any nontrivial linear equation has a constant fraction of solutions that are monochromatic in any $2$-coloring of…
A system of linear forms $L=\{L_1,\ldots,L_m\}$ over $\mathbb{F}_q$ is said to be Sidorenko if the number of solutions to $L=0$ in any $A \subseteq \mathbb{F}_{q}^n$ is asymptotically as $n\to\infty$ at least the expected number of…
We prove that suitably generic pairs of linear equations on an even number of variables are uncommon. This verifies a conjecture of Kam\v{c}ev, Morrison and the second author. Moreover, we prove that any large system containing such a…
A system of linear equations $L$ is said to be norming if a natural functional $t_L(\cdot)$ giving a weighted count for the set of solutions to the system can be used to define a norm on the space of real-valued functions on…
A classical question in combinatorial number theory asks whether an equation has a solution inside a particular subset of its domain. The Rado's Theorem gives a set of necessary and sufficient conditions for a systems of linear equations to…
A linear equation is $r$-regular, if, for every $r$-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 $x_1 + 2x_2 + \cdots…
A graph $H$ is said to be common if the number of monochromatic labelled copies of $H$ in a red/blue edge colouring of a large complete graph is asymptotically minimized by a random colouring with an equal proportion of each colour. We…
A graph $H$ is common if the limit as $n\to\infty$ of the minimum density of monochromatic labelled copies of $H$ in an edge colouring of $K_n$ with red and blue is attained by a sequence of quasirandom colourings. We apply an…
Parametric linear systems are linear systems of equations in which some symbolic parameters, that is, symbols that are not considered to be candidates for elimination or solution in the course of analyzing the problem, appear in the…
A graph $H$ is \emph{common} if the number of monochromatic copies of $H$ in a 2-edge-colouring of the complete graph $K_n$ is asymptotically minimised by the random colouring, or equivalently, $t_H(W)+t_H(1-W)\geq 2^{1-e(H)}$ holds for…
We introduce a new class of "random" subsets of natural numbers, WM sets. This class contains normal sets (sets whose characteristic function is a normal binary sequence). We establish necessary and sufficient conditions for solvability of…
Differential systems with a Fuchsian linear part are studied in regions including all the singularities in the complex plane of these equations. Such systems are not necessarily analytically equivalent to their linear part (they are not…
A linear system of plane curves satisfying multiplicity conditions at points in general position is called special if the dimension is larger than the expected dimension. A (-1) curve is an irreducible curve with self intersection -1 and…