Intereting Posts

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calculate the sum of an infinite series
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As you can see I am woring at three excercises:

Prove that following classes for graphs are non-definiable (on FO sentence):

a) biparted graphs

b) even number of nodes

c) graphs contains eulerian cycle

a) We know that biparted graphs can’t contain odd cycle. So lets $n$ will be number of rounds. Now, Let $A$ will be cycle (biparted graph) of length $2^{n}+2$ and $B$ will be cycle of length $2^n+1$. We know, that in $n$ rounds duplicator has winning stratrgy.

b) Similar apporach as in (a). Two chains of lenght $2^n+2$ and $2^n+1$.

c) This one seems to be the hardest. I think that we can give two structures for $n$ rounds:

$A$ is cycle of length $2^{n+1}$ and $B$ is chain of length $2^{n+1}$

Am I ok ?

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- Understanding common knowledge in logic and game theory

(a) Construct a set $S$ of axioms saying “$G$ is not bipartite and every vertex in $G$ has exactly two neighbours.” What graphs satisfy $S$? Now add a set $T$ of axioms that force $G$ to have no cycles, and use compactness on $S \cup T$.

(a*) Note that “$G$ is bipartite” is axiomatizable. Why? And where does the proof of (a) fail to show that it is not axiomatizable?

(b) Even implies finite. So following the same method described here leads to a proof.

(c) Eulerian cycle also implies finite. So same as (b).

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