4 research outputs found
On First-Order Definable Colorings
We address the problem of characterizing -coloring problems that are
first-order definable on a fixed class of relational structures. In this
context, we give several characterizations of a homomorphism dualities arising
in a class of structure
C4-free subgraphs with large average degree
Motivated by a longstanding conjecture of Thomassen, we study how large the
average degree of a graph needs to be to imply that it contains a -free
subgraph with average degree at least . K\"uhn and Osthus showed that an
average degree bound which is double exponential in t is sufficient. We give a
short proof of this bound, before reducing it to a single exponential. That is,
we show that any graph with average degree at least (for
some constant ) contains a -free subgraph with average degree at
least . Finally, we give a construction which improves the lower bound for
this problem, showing that this initial average degree must be at least