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Is there a "Cauchy-Schwarz proof" of the following inequality? Theorem. Given $f \colon [0,1]^2 \to [0,1]$, one has $$ \int_{[0,1]^4} f(x,y)f(z,y)f(z,w) \, dxdydzdw \geq \left(\int_{[0,1]^2} f(x,y) \,...

I guess a discrete-mathematics-related question is still welcome in MO since I was new to the community and learned from this amazing past post. The following claim is a simplified and abstract form ...

It is known that for triangle-free graphs, if they are $d$-regular, then $2d\leq n$, where $n$ is the number of vertices. In words, the degree is less than or equal to half the number of vertices (...

Consider a family $\mathcal{G} = \{ A_1,B_1,\ldots,B_m \}$ of $m+1$ non-empty finite distinct sets with the following property: $$A_1 \cap B_k = \emptyset, 1 \le k \le m$$ Let $\mathcal{F} = \{A_1 \...