Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: This is a continuation of our work on quasi-random graph properties. The class of quasi-random graphs is defined by certain equivalent graph properties possessed by random graphs. One of the most important of these properties is that, for fixed $\nu$, every fixed sample graph $L_\nu$ has the same frequency in $G_n$ as in the $p$-random graph. (This holds for both induced and not necessarily induced containment.) In &#005B;9&#005D;  proved that, if the frequency of just one fixed $L_\nu$ – as a not necessarily induced subgraph – in every ‘large’ induced subgraph $F_h\subseteq G_n$ is the same as for the random graphs, then $(G_n)$ is quasi-random. Here  shall investigate the analogous problem for induced subgraphs $L_\nu$. In such cases $(G_n)$ is not necessarily quasi-random. shall prove, among other things, that, for every regular sample graph $L_\nu$, $\nu\geqslant 4$, if the number of induced copies of $L_\nu$ in every induced $F_h\subseteq G_n$ is asymptotically the same as in a $p$-random graph (up to an error term $o(n^\nu)$), then $(G_n)$ is the union of (at most) two quasi-random graph sequences, with possibly distinct attached probabilities (assuming that $p\in (0,1)$, $e(L_\nu)>0$, and $L_\nu\ne K_\nu$). conjecture the same conclusion for every $L_\nu$ with $\nu\ge 4$, i.e., even if  drop the assumption of regularity. shall reduce the general problem to solving a system of polynomials. This gives a ‘simple’ algorithm to decide the problem for every given $L_\nu$.
['Miklós Simonovits', 'Vera T. Sós']
I am writing a research paper on a idea Hereditary Extended Properties, Quasi-Random Graphs and Induced Subgraphs
[{"from": "human", "value": "I am writing a research paper on a idea Hereditary Extended Properties, Quasi-Random Graphs and Induced Subgraphs"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: This is a continuation of our work on quasi-random graph properties. The class of quasi-random graphs is defined by certain equivalent graph properties possessed by random graphs. One of the most important of these properties is that, for fixed $\\nu$, every fixed sample graph $L_\\nu$ has the same frequency in $G_n$ as in the $p$-random graph. (This holds for both induced and not necessarily induced containment.) In &#005B;9&#005D;  proved that, if the frequency of just one fixed $L_\\nu$ \u2013 as a not necessarily induced subgraph \u2013 in every \u2018large\u2019 induced subgraph $F_h\\subseteq G_n$ is the same as for the random graphs, then $(G_n)$ is quasi-random. Here  shall investigate the analogous problem for induced subgraphs $L_\\nu$. In such cases $(G_n)$ is not necessarily quasi-random. shall prove, among other things, that, for every regular sample graph $L_\\nu$, $\\nu\\geqslant 4$, if the number of induced copies of $L_\\nu$ in every induced $F_h\\subseteq G_n$ is asymptotically the same as in a $p$-random graph (up to an error term $o(n^\\nu)$), then $(G_n)$ is the union of (at most) two quasi-random graph sequences, with possibly distinct attached probabilities (assuming that $p\\in (0,1)$, $e(L_\\nu)>0$, and $L_\\nu\\ne K_\\nu$). conjecture the same conclusion for every $L_\\nu$ with $\\nu\\ge 4$, i.e., even if  drop the assumption of regularity. shall reduce the general problem to solving a system of polynomials. This gives a \u2018simple\u2019 algorithm to decide the problem for every given $L_\\nu$."}]