Partition models, Permutations of infinite sets without fixed points, Variants of CAC, and weak forms of AC

We study new relations of the following statements with weak choice principles in ZF and ZFA. 1. For every infinite set X, there exists a permutation of X without fixed points. 2. There is no Hausdorff space X such that every infinite subset of X contains an infinite compact subset. 3. If a field has an algebraic closure then it is unique up to isomorphism. 4. Variants of Chain/Antichain principle. 5. Any infinite locally finite connected graph has a spanning subgraph omitting some complete bipartite graphs. 6. Any infinite locally finite connected graph has a spanning m bush for any even integer m greater than 4. We also study the new status of different weak choice principles in the finite partition model (a type of permutation model) introduced by Bruce in 2016. Further, we prove that Van Douwens Choice Principle holds in two recently constructed known permutation models.