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I am given a graph G with 1000 vertices and 250,001 edges and I need to show there are two triangles in it with common edge.

Now if I write the graph of 'two triangles with a common edge' as H then I need to show that:

[tex]ex(1000,H) \le ex(1000, K_3) = 250,000[/tex]

So basically I need to show that the set

[tex] F_1=\{ |E(G)| : |V(G)|=1000, H\not\subset G\}[/tex]

is a subset of [tex] F_2=\{|E(G)| : |V(G)|=1000, K_3 \not\subset G\}[/tex]

Now, I am not sure if my reasoning is correct, but I think that if K_3 isn't a subgraph of G then obviously H isn't as well (cause G doesn't contain triangles), so it seems to me that F_2 is a subset of F_1 and not the other way around?

Have I got it all wrong here?

Now if I write the graph of 'two triangles with a common edge' as H then I need to show that:

[tex]ex(1000,H) \le ex(1000, K_3) = 250,000[/tex]

So basically I need to show that the set

[tex] F_1=\{ |E(G)| : |V(G)|=1000, H\not\subset G\}[/tex]

is a subset of [tex] F_2=\{|E(G)| : |V(G)|=1000, K_3 \not\subset G\}[/tex]

Now, I am not sure if my reasoning is correct, but I think that if K_3 isn't a subgraph of G then obviously H isn't as well (cause G doesn't contain triangles), so it seems to me that F_2 is a subset of F_1 and not the other way around?

Have I got it all wrong here?

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