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m Replace "peculiarity" with "weakness" in regards to nonconstructivity. Nonconstructive proofs are common in mathematics, so not particularly peculiar, but are weaker than constructive proofs, as described in the following sentence. |
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:<math>E[X(S_r)] = {n \choose r}2^{1-{r \choose 2}}.</math>
Consider what happens if this value is less than {{math|1}}. Since the expected number of monochromatic {{mvar|r}}-subgraphs is strictly less than {{math|1}}, it must be that a specific random coloring satisfies the condition that the number of monochromatic {{mvar|r}}-subgraphs is strictly less than {{math|1}}. The number of monochromatic {{mvar|r}}-subgraphs in this random coloring is a non-negative integer, hence it must be {{math|0}} ({{math|0}} is the only non-negative integer less than {{math|1}}). It follows that if
:<math>E[X(S_r)] = {n \choose r}2^{1-{r \choose 2}} < 1</math>
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