Sparse binary data is just a term for when you have many variables and when you break them down there are very few or no persons in the subgroupings.

Variables: Age, Sex, Exposure, Race, Insurance status, marital status, employment status,..., etc. So if you have 10 outcomes in the smaller of the two outcome groups, well if all of the variables were binary there would be 128 subgrouping combinations of these variables though you only have 10 people with the lesser outcome, so most of the combinations will be empty (e.g., no young, males with exposure, asian, insured, unmarried, and unemployed - thus it becomes goofy making predictions about people you don't even have data for even though created beta coefficients. Also, models have troubles converging in these scenarios. Does that make sense?

Is your dependent variable actually binary? Let us say it is, so your DV has two groups we will call 0 or 1 (which could represent Yes and No). So say 33 people are 1s and 10 are 0s. the smaller group is '0'. So a general rule is you need 10 - 20 people in your smaller group for each predictor you introduce to your model, So for 1 IV you would need 10-20 people, 2 IVs 20-40,..., 6 IVs (what you have) 60-120 people with 0s, so overall you would approximately need 6*(10 for smaller group), which would also mean 6*33 by default, so moreover your n=43 times 6-10, so a sample of 258 just to meet the bottom threshold of the general rule. The rule just gives you a generic marker to think about. Now if you added an interaction term that would mean you need to add at least another 43 people overall so the lesser group would increase by 10. I wrote this quickly, so I apologize for typos and clarity.

Your understanding of the project's context should guide you in what IVs to use, so it is not just a fishing expedition, where you say hey that came up significant. In addition, if there are too many predictors in the model you run the risk of over-fitting data using a small sample, which means your results may not be generalizable to other samples, they are too case specific, and your case (sample) is just one realization of the true population, so given sampling variability the next sample could differ.

So you can limit the number of predictors you try or perhaps use a penalization (Firth) or regularization model (Lasso, E-net), which the latter will get rid of correlated predictors and whittle your candidate set down to something the model can support.