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I do not understand the proof of Bollobas and Thomason of an embedding lemma. There is a lot of notation to present first, then the statement of the lemma, then the precise question about the proof.

Let $U$ and $W$ be non-empty sets of vertices of a graph $G$. Let $e(U,W)$ be the number of edges with one endpoint in $U$ and the other in $W$. Let $d(U,W)=\frac{e(U,W)}{\lvert U\rvert\lvert W\rvert}$. The pair $(U,W)$ is $\eta$-uniform if $\lvert d(U,W)-d(U',W')\rvert<\eta$ whenever $\emptyset\ne U'\subseteq U$ and $\emptyset\ne W'\subseteq W$ and $\lvert U'\rvert>\eta\lvert U\rvert$ and $\lvert W'\rvert>\eta\lvert W\rvert$.

Lemma 3 (from Bollobas and Thomason, "Hereditary and Monotone Properties of Graphs" in The Mathematics of Paul Erdős II). Let $H$ be a graph with vertex set $\{x_1,\dots,x_k\}$. Let $0<\lambda,\eta<1$ satisfy $k\eta\le\lambda^{k-1}$. Let $G$ be a graph with vertex set $\bigcup_{i=1}^k V_i$ where the $V_i$ are disjoint sets each of order $u\ge 1$. Suppose that each pair $(V_i,V_j)$ is $\eta$-uniform and that $d(V_i,V_j)\le 1-\lambda$ if $x_i x_j\notin E(H)$ and that $d(V_i,V_j)\ge \lambda$ if $x_i x_j\in E(H)$.

Then there exist vertices $v_i\in V_i$ such that the map $x_i\mapsto v_i$ gives an isomorphism between $H$ and the subgraph of $G$ spanned by $\{v_1,\dots,v_k\}$.

In the proof, they say they may assume $H$ is a complete graph. Why?

I have a follow-up question that I will put in a new post.

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  • $\begingroup$ By the way, I mention because I didn't know for the longest time: it is not Erdös but Erdős. I have edited accordingly. $\endgroup$ – LSpice May 20 at 1:47
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    $\begingroup$ Thanks. (For that matter, his name isn't "Paul.") Alert Springer Verlag, because I literally copied the title from the Springer Verlag site. On the cover of the book they used the "two dots" umlaut as well---a book co-edited by Ron Graham. link.springer.com/book/10.1007/978-3-642-60406-5 media.springernature.com/w306/springer-static/cover-hires/book/… (There is a different cover where they use the other mark.) So here's a question for MLA: if a book misspells a word, and one refers to the title of the book, should one correct the misspelling? $\endgroup$ – Tri May 20 at 5:44
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The reduction here is: if $x_ix_j$ is not an edge of $H$, then swap edges and non-edges between $V_i$ and $V_j$. Looking for a complete graph in the result, with vertices in specified parts, is the same as looking for a copy of $H$ in the original graph, and the uniformity is preserved.

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