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This paper is the fourth and the last part of the series "Spherical higherorder Fourier analysis over finite fields", aiming to develop the higher orderFourier analysis method along spheres over finite fields, and to solve theGeometric Ramsey Conjecture in the finite field setting.  In this paper, we proof a conjecture of Graham on the Remsey properties forspherical configurations in the finite field setting. To be more precise, weshow that for any spherical configuration $X$ of $\mathbb{F}_{p}^{d}$ ofcomplexity at most $C$ with $d$ being sufficiently large with respect to $C$and $\vert X\vert$, and for some prime $p$ being sufficiently large withrespect to $C$, $\vert X\vert$ and $\epsilon>0$, any set $E\subseteq\mathbb{F}_{p}^{d}$ with $\vert E\vert>\epsilon p^{d}$ contains at least$\gg_{C,\epsilon,\vert X\vert}p^{(k+1)d-(k+1)k/2}$ congruent copies of $X$,where $k$ is the dimension of $\text{span}_{\mathbb{F}_{p}}(X-X)$. The noveltyof our approach is that we avoid the use of harmonic analysis, and replace itby the theory of spherical higher order Fourier analysis developed in previousparts of the series.

Spherical higher order Fourier analysis over finite fields IV: an application to the Geometric Ramsey Conjecture