Ryser Type Conditions for Extending Colorings of Triples

Abstract

In 1951, Ryser showed that an n ˆn array L whose top left r ˆs subarray is filled with n different symbols, each occurring at most once in each row and at most once in each column, can be completed to a latin square of order n if and only if the number of occurrences of each symbol in L is at least r `s ´n. We prove a Ryser type result on extending partial coloring of 3-uniform hypergraphs. Let X, Y be finite sets with X Ĺ Y and |Y | " 0 pmod 3q. When can we extend a (proper) coloring of λ `X 3 ˘(all triples on a ground set X, each one being repeated λ times) to a coloring of λ `Y 3 ˘using the fewest possible number of colors? It is necessary that the number of triples of each color in `X 3 ˘is at least |X| ´2|Y |{3. Using hypergraph detachments (Combin. Probab. Comput. 21 (2012), 483-495), we establish a necessary and sufficient condition in terms of list coloring complete multigraphs. Using Häggkvist-Janssen's bound (Combin. Probab. Comput. 6 (1997), 295-313), we show that the number of triples of each color being at least |X|{2 ´|Y |{6 is sufficient. Finally we prove an Evans type result by showing that if |Y | ě 3|X|, then any q-coloring of any subset of λ `X 3 ˘can be embedded into a λ `|Y |´1 2 ˘-coloring of λ `Y 3 ˘as long as q ď λ `|Y |´1 2 ˘´λ `|X| 3 ˘{ t|X|{3u.