3. The Galvin-Hajnal theorems.

In this section I want to present two theorems of Galvin and Hajnal that greatly generalize Silver’s theorem. I focus on a “*pointwise*” (or *everywhere*) result, that gives us information beyond the pointwise theorems from last lecture, like Corollary 23. Then I state a result where the hypotheses, as in Silver’s theorem, are required to hold* stationaril*y rather than everywhere. From this result, the full version of Silver’s result can be recovered.

Both results appear in the paper Fred Galvin, András Hajnal, *Inequalities for Cardinal Powers*, The Annals of Mathematics, Second Series, 101 (3), (May, 1975), 491–498, available from JSTOR, that I will follow closely. For the notion of -inaccessibility, see Definition II.2.20 from last lecture.

**Theorem 1.** *Let be uncountable regular cardinals, and suppose that is -inaccessible. Let be a sequence of cardinals such that for all Then also *

The second theorem will be stated next lecture. Theorem 1 is a rather general result; here are some corollaries that illustrate its reach:

**Corollary 2.** *Suppose that are uncountable regular cardinals, and that is -inaccessible. Let be a cardinal, and suppose that for all cardinals Then also *

**Proof. **Apply Theorem 1 with for all

**Corollary 3. ***Suppose that are uncountable regular cardinals, and that is -inaccessible. Let be a cardinal of cofinality and suppose that for all cardinals Then also *

**Proof. **Let be a sequence of cardinals smaller than such that and set for all Then for all by assumption. By Theorem 1, as well.

**Corollary 4.** *Let be cardinals, with and regular and uncountable. Suppose that for all cardinals Then also *

**Proof.** This follows directly from Corollary 2, since is regular and -inaccessible.

**Corollary 5. ***Let be cardinals, with and of uncountable cofinality Suppose that for all cardinals Then also *

**Proof. **This follows directly from Corollary 3 with

**Corollary 6.** *Let be an ordinal of uncountable cofinality, and suppose that for all Then also *

**Proof.** This follows from Corollary 5 with and

**Corollary 7. ***Let be an ordinal of uncountable cofinality, and suppose that for all cardinals and all Then also *

**Proof.** This follows from Corollary 4: If , then by Theorem II.1.10 from lecture II.2. But so both and are strictly smaller than

**Corollary 8.*** If for all then also *

**Proof.** By Corollary 5.

**Corollary 9.** *If for all then also *

**Proof.** By Corollary 7.

Notice that, as general as these results are, they do not provide us with a bound for the size of for the first cardinal of uncountable cofinality that is a fixed point of the aleph sequence, not even under the assumption that is a strong limit cardinal.