Polynomial removal lemmas for ordered graphs

A recent result of Alon, Ben-Eliezer and Fischer establishes an induced removal lemma for ordered graphs. That is, if \(F\) is graph \(\varepsilon›0\), then there exists \(\delta_{F}(\varepsilon)›0\) such that every \(n\)-vertex \(G\) containing at most \(\delta_{F}(\varepsilon) n^{v(F)}\) copies can be made \(F\)-free by adding/deleting \(\varepsilon n^2\) edges. We prove \(\delta_{F}(\varepsilon)\) chosen to a polynomial function \(\varepsilon\) only \(|V(F)|=2\), or the with vertices \(x‹y‹z\) edges \(\{x,y\},\{x,z\}\) (up complementation reversing vertex order). also discuss similar problems in non-induced case.Mathematics Subject Classifications: 05C35, 05C75Keywords: Ordered graph,