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→Sidorenko graphs are common: included citations and evidence of truthfulness of the proofs. |
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\ge 2 \bigg( \frac{t(K_2, W) + t(K_2, 1 - W)}{2} \bigg)^{e(F)} = 2^{-e(F) + 1}</math>
Thus, the conditions for common graph is met. This proof, along with proof of <math>K_4^{-}</math> being common graph, can be seen from page 297-298 of the aforementioned book of László Lovász<ref>{{Cite book|last=Lovász|first=László|title=Large Networks and Graph Limits|publisher=American Mathematical Society Colloquium publications|year=2012|isbn=978-0821890851|___location=United States|pages=297-298|language=English}}</ref>.
===The triangle graph is common===
Line 74:
<math>t(K_3, W) + t(K_3, 1 - W) \ge 1 - 3 t(K_2, W) + 3 t(K_{2}, W)^2
= 1/4 + 3 \big( t(K_2, W) - 1/2 \big)^2 \ge 1/4</math>.
This above proof can be obtained from taking continuous analog of Theorem 1 in Goodman 1959 paper, "On Sets Of Acquaintances And Strangers At Any Party"<ref>{{Cite journal|last=Goodman|first=A. W.|date=1959|title=On Sets of Acquaintances and Strangers at any Party|url=https://www.jstor.org/stable/2310464|journal=The American Mathematical Monthly|volume=66|issue=9|pages=778–783|doi=10.2307/2310464|issn=0002-9890}}</ref>.
== See also ==
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