1. Finite Model Theory Lecture 19
Summary on 0/1 Laws

2. Random Graphs G(n, p): each edge (i,j) has probability p
Think of p as p(n)
Examples:
p = 1/2 Fagin’s framework
p = S/n2 expected graph size = S
p = 1/n3/2
Etc.

3. Evolution
Erdos and Reny: evolution of random graphs, as p(n) “grows” from 0 to 1
Spencer, Lynch, and others: evolution of FO sentences as p(n) “grows” from 0 to 1
Will discuss next, following J. Spencer … … … 1
p(n)

4. Proving a 0/1 Law Suppose a 0/1 law holds for some p(n)
Let T1 = { f | limn Pr(f) = 1}
T1 is a complete theory: 8 f. f 2 T1 or : f 2 T1 [ WHY ??]
Hence: 0/1 law holds iff there exists a theory T s.t.
Th(T) = { f | T |= f} is complete (e.g w-categorical)
8 f 2 T, limn Pr(f) = 1 (here we use the particular p(n))

5. 1. The Void p(n) ¿ 1/n2 Expected graph size: p(n) £ n2 ! 0
Hence limn Pr(G= ;) = 1
What is the theory T here ?
T = { Ø$ x.$ y.R(x,y) }

6. 2. On the k’th Day… 1/n1+1/(k-1) ¿ p(n) ¿ 1/n1+1/k
Let’s try some examples, for k = 4:
1/n4/3 ¿ p(n) ¿ 1/n5/4 1
H1 =
Limn Pr(H1) =
H2 =
Limn Pr(H2) =
H3 =
Limn Pr(H3) =
H4 =
Limn Pr(H4) =

7. 2. On the k’th Day… The theory T: There are no cycles
For every tree H with · k edges and 8 r >0 H appears as a CC at least r times
There are no trees H with k+1 edges
: 9 x1…9 xk. R(x1,x2)Æ…ÆR(xk,x1)
T is w categorical [ WHY ?? WHAT IS R HERE ?? ]

8. 3. On Day w 1/n1+e ¿ p(n) ¿ 1/n, 8 e Example: p(n) = 1/(n ln(n))
The theory T is:
There are no cycles
For every finite tree H and 8 r > 0 H appears as a cc component at least r times
T is NOT w categorical [ WHY ?? ] Still, it is complete [ WHY ?]

9. 4. Past the Double Jump 1/n ¿ p(n) ¿ ln(n)/n The theory T:
8 k, there are no k nodes with ¸ k+1 edges
8 k ¸ 3, 8 r: there are ¸ r copies of the cycle Ck
8 k ¸ 3, s, d: there is no Ck and a vertex of degree d at distance s from Ck
For every tree H and 8 r > 0 H appears as a cc at least r times
Again, T is not w-categorical, but still complete

10. 5. Beyond Connectivity Ln(n)/n ¿ p(n) ¿ 1/n1-e, 8 e
Now the random graph G(n,p) becomes connected
The theory T is:
8 k: there are no k nodes with ¸ k+1 edges
For every d: all vertices have at least d neighbors
For every r and k ¸ 3: there exists at least r copies of Ck
And, again, T is not w-categorical, but still complete

11. 6. a Irrational P(n) = 1/na, a 2 (0,1) irrational
Then FO has a 0/1 law
The theory T is:
For every graph H with v vertices e edges s.t. v < e a, there does not exists a copy of H
“Generic extension axioms” There are complicated, but mimic extension axioms.

12. 7. a Rational
P(n) = 1/na, a 2 (0,1) irrational
Then FO has no 0/1 law