**Ondřej Ježil**
### Pseudofinite structures and limits

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Charles University, Prague, 2022

**Abstract**
For a class of graph instances of a computational problem we define a
limit object, relative to some computationally restricted class of functions. The
key method here is forcing with random variables where the sample set is taken as
instances of some nonstandard size. We study the general theory of these limits,
called in the thesis wide limits, and their connection to classical problems such
as finding a large clique or with the combinatorial problems associated with the
classes of total NP search problems PPA and PPAD. Our main results are
several 0-1 laws associated with these limits and existence of a significantly large
clique of the wide limit of all graph consisting of one large clique.

**Table of Contents **
Introduction (p. 2)
Preliminaries (p. 3)
The ambient model M (p. 3)
Nonstandard analysis (p. 3)
Total NP search problems and polynomial oracle time (p. 4)
1 Forcing with random variables and the wide limit (p. 6)
1.1 Setup (p. 6)
1.2 The first order wide limit (p. 7)
1.3 The second order wide limit (p. 8)
1.4 The vertex family Frud and Grud (p. 9)
1.5 Different choices of n (p. 10)
1.6 Theories of wide limits (p. 11)
2 General theory (p. 12)
2.1 \mathcal{G}_k = \text{EDGE}_k (p. 12)
2.2 Sparse \mathcal{G}_k (p. 13)
2.3 Dense \mathcal{G}_k (p. 16)
2.4 Gk = \text{ALL}_k (p. 19)
2.5 Isomorphism closed categorical \mathcal{G}_k (p. 22)
3 Dense case (p. 24)
3.1 Gk = SKk (p. 24)
3.2 Gk = CKk (p. 27)
4 Sparse case and TFNP (p. 29)
4.1 Gk = ∗PATH_k (p. 29)
4.2 Gk = ∗PATH^≤_k (p. 33)
4.3 Gk = ∗DPATH_k (p. 35)
Concluding remarks (p. 37)
Bibliography (p. 38)