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You might consider appealing to computationally intractable problems that can be wrangled by mathematical bounding. This could double as an argument against those students who might say, "Why bother learning math? Can't I just make my computer solve things for me?"

I believe that students of a variety of ages can understand an argument that $R(3,3)=6$. Depending on whether you're talking to high-schoolers or undergrad math majors, you might phrase the setup of the problem differently, or go into different depths in the argument. But they can all follow along.

Follow this up with asking about $R(4,4)$. If they can't find the exact number "by hand", could they program a computer to do it? Work with them to realize how many possible 2-colorings of $K_n$ there are, and estimate how long it would take to check one graph for a monochromatic $K_4$. (You can be hand-wavey here, depending on the audience.)

Continue for $R(5,5)$ and $R(6,6)$. Try to explain how the computational time required explodes. Explain how we have no hope of simply letting a computer run to figure out an answer. By proving better and better bounds, we can narrow the search space, but without doing that, we're essentially knowledge-less.

Then, hit 'em with this quote (idea attributed to Erdős, words by Joel Spencer):