A partition regular system is a linear system of equations of the form $A\cdot x=0$, which satisfies a Ramsey-type result (namely, that for each $r>0$ whenever we colour the integers in $r$ classes, there is a class which contains a monochromatic solution). The well-known Rado's theorem gives a characterization of such matrices, but we are not interested on this here.


An strong condition of this notion is density regular: a matrix $A$ is density regular if for every $\varepsilon>0$ and $Y\subset \{1,\dots,n\}$ with $|Y| \geq \varepsilon n$, then $Y$ contains a solution to the equation $A\cdot x=0$. This is the strong counterpart of Rado's theorem, and it was proven by Frankl, Rödl and Graham that $A$ is density regular iff the vector $x=(1,\dots, 1)$ is a solution of the system (namely, the columns vectors of $A$ sum 0). For instance, the matrix equation associated to $k$-APs satisfies this condition, so Szemerédi's theorem is covered by this result.


After this, it comes my question. Take the Schur equation (x+y=z), which is partition regular but NOT density regular. Easily, there are sets of linear size (for instance, take the odd numbers) which are solution-free. However, it is very easy to show that if $X\in \{1,\dots,n\}$ satisfies $|X|\geq \left({\frac{1}{2}+\varepsilon}\right) n$, then $X$ contains a Schur triple.


My question is the following: is a similar result true for general partition regular systems? In other words, is it true the following statement?: let $A$ be a partition regular system. Then there exists $C:=C(A)<1$ such that for $n$ large enough every subset $X\subset \{1,\dots, n\}$ with $|X|> C n$ contains a solution to the equation $A \cdot x=0$. 



Partition regular systems: do they have solution in (very dense) set of integers?
This doesn't have anything to do with partition regularity: There is such a constant $C(A)<1$ provided only that there exists at least one solution to $Ax=0$ in positive integers.


Indeed suppose $x = (x\_1,\dots,x\_m)$ is a solution. Then $jx = (jx\_1,\dots,jx\_m)$ is a solution for each $j\geq 1$. Now take a large integer $n$ much larger than $m$ and $\max x\_i$ and allow $j$ to range between $1$ and $J = \lfloor n/\max x\_i\rfloor $. Every element of $\{1,\dots,n\}$ appears in at most $m$ of the solutions $x,2x,\dots,Jx$, so if none of $x,2x,\dots,Jx$ is wholly contained in our set $X$ then $X$ is missing at least $J/m$ elements, so $$|X| \leq \left(1-\frac{1}{m \max x\_i}\right) n + 1.$$


