1. **Problem statement:** Determine if there exists a standard graded algebra $R$ such that $\dim R_1=4$ and $\dim R_3=3$. 2. **Recall definitions:** A standard graded algebra $R=\bigoplus_{n\geq0} R_n$ over a field satisfies $R_0=k$ (a field) and is generated by $R_1$ as an algebra. 3. **Key property:** For a standard graded algebra generated in degree 1, the dimension of $R_n$ is at least the dimension of the $n$-th symmetric power of $R_1$, unless relations reduce it. 4. **Dimension of symmetric powers:** Since $\dim R_1=4$, the dimension of the degree 3 part of the polynomial ring in 4 variables is $$\dim S^3(R_1) = \binom{4+3-1}{3} = \binom{6}{3} = 20.$$ 5. **Relations:** To have $\dim R_3=3$, many relations must reduce the dimension from 20 to 3 in degree 3. 6. **Hilbert function constraints:** The Hilbert function $h(n) = \dim R_n$ of a standard graded algebra generated in degree 1 is nondecreasing for small $n$ and typically grows or stabilizes. 7. **Contradiction:** Here, $h(1)=4$ but $h(3)=3 < 4$, which violates the growth condition of Hilbert functions of standard graded algebras. 8. **Conclusion:** No standard graded algebra $R$ exists with $\dim R_1=4$ and $\dim R_3=3$ because the Hilbert function cannot decrease from 4 to 3 in degree 3. **Final answer:** No such standard graded algebra exists.