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During lectures I came across following statement:

If you want Hosmer-Lemeshow test to be valid, number of expected events ($E_1g$) should be >5 in most of $g$ groups

Then after few lectures, lecturer said:

If you want Hosmer-Lemeshow test to be valid, number of expected events($E_1g$) and non-events ($E_0g$) should be >5 in 80% of cells.

I googled whole internet, read 3 most popular books about logistic regression and still can find answer about which version is correct.

The question is: which recommendation for Hosmer-Lemeshow test is proper one? Or maybe you know another one and can give some papers about it?

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Hosmer and Lemeshow answered this question in 1980 in their paper Sample size requirements for studies estimating odds ratios or relative risks to be find here. In further works they refer to this paper.

The values stated in your lecture refer to the recommendations by William G. Cochran, first published 1954 in his paper Some Methods for Strengthening the Common $\chi^2$ Tests. His works refers to the pearson-chi-squared-test. These assumptations are cited often for any chi-squared statistics.

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