Partition regular systems: do they have solution in (very dense) set of integers? 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 $A \cdot x=0$. 
 A: 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 $$|X| \leq \left(1-\frac{1}{m \max x_i}\right) n + 1.$$
A: Yes, this statement is true. If you read Rado's proof (or any of the standard proofs) you can modify it to obtain a 'Ramsey multiplicity' type statement, which says that there is not just one monochromatic solution to any PR system in an $r$-colouring of $[n]$, but actually there is $\delta>0$ (depending on the system!) such that a $\delta$-fraction of all the solutions in $[n]$ are monochromatic. (Note: this is a fair bit of work.)
Now, since no integer in $[n]$ is in much more than the average number of solutions (easy to check, constant again depending on the PR system), however you remove $\ll\delta n$ integers there will remain solutions.
The constant you get from this argument is surely very far from the truth - it would be an interesting problem to try to find the optimal constant for general PR systems, but I would guess this is hard.
