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We determine Turán numbers for the family of 3‐uniform minimal paths of length four for all . We also establish the second‐ and third‐order Turán numbers and use them to compute the corresponding Ramsey numbers for up to four colors.
For two given graphs G and H the planar Ramsey number PR(G,H) is the smallest integer n such that every planar graph F on n vertices either contains a copy of G or its complement contains a copy H. By studying the existence of subhamiltonian cycles in complements of sparse graphs, we determine all planar Ramsey numbers for pairs of cycles.
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