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Removed the "thank you" which made this seem like a non-answer.
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Stefan Kohl
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Thanks KP for the write up! One small comment,remark on KP Hart's answer: if one does the same to a finitely branching tree with unbounded sizes of the splitting sets then one gets a MAD family with no infinite delta-system but delta systems of all finite sizes.

Thanks KP for the write up! One small comment, if one does the same to a finitely branching tree with unbounded sizes of the splitting sets then one gets a MAD family with no infinite delta-system but delta systems of all finite sizes.

One remark on KP Hart's answer: if one does the same to a finitely branching tree with unbounded sizes of the splitting sets then one gets a MAD family with no infinite delta-system but delta systems of all finite sizes.

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Thanks KP for the write up! One small comment, if one does the same to a finitely branching tree with unbounded sizes of the splitting sets then one gets a MAD family with no infinite delta-system but delta systems of all finite sizes.