To solve the given problem, we need to find the smallest non-negative integer $n$ such that any monomial $x^i y^j z^k$ with $i + j + k \geq n$ can be expressed in the form:

$$(x + y + z)P(x, y, z) + (xy + yz + zx)Q(x, y, z) + xyzR(x, y, z)$$

where $P, Q, R$ are polynomials with integer coefficients.

### Step-by-step Analysis

1. Understanding the Problem:
- The monomial $x^i y^j z^k$ needs to be expressed as a polynomial that results from the specific linear combination given in the problem.
- We need to analyze the degrees that can be formed by $(x+y+z)P$, $(xy+yz+zx)Q$, and $xyzR$.

2. Degrees of Terms:
- The term $(x + y + z)P$ contributes degree $\deg(P) + 1$.
- The term $(xy + yz + zx)Q$ contributes degree $\deg(Q) + 2$.
- The term $xyzR$ contributes degree $\deg(R) + 3$.

3. Constructing a Basis for High Degrees:
- For $x^i y^j z^k$ with $i + j + k$ sufficiently large, study the combinations of terms that can sum to this degree.
- Notice that:
- $(x+y+z)x^{i-1}y^jz^k$ produces monomials like $x^iy^jz^k$, $x^{i-1}y^{j+1}z^k$, and $x^{i-1}y^jz^{k+1}$.
- $(xy+yz+zx)x^{i-1}y^{j-1}z^k$ produces monomials like $x^iy^jz^k$, $x^{i-1}y^{j+1}z^{k+1}$, etc.
- $xyzx^{i-1}y^{j-1}z^{k-1}$ directly gives $x^iy^jz^k$.

4. **Inferring the Value of $n$:**
- Observe that for $i + j + k = 3$, the simplest monomial expressions such as $x^3, y^3, z^3$ can't be formed using any combination of the terms, as these require linear alternation terms which can't have degree less than 3.
- Once $i + j + k \geq 4$, every required monomial can be constructed using the given forms by expressing simpler terms and adding higher degree components systematically using $P, Q, R$.

5. Conclusion:
- The construction of every monomial becomes feasible for $i + j + k \geq 4$. Therefore, the smallest $n$ for which each monomial in $x^i y^j z^k$ can be expressed in the form of the given polynomial combination is:

$$\boxed{4}$$

This reasoning shows that once the total degree $i + j + k$ reaches 4, $x^i y^j z^k \in \mathcal{B}$, validating $n = 4$ as the smallest such integer.