Horses in warfare of today's English saddle. Both the Scythians and Assyrians used pads with added felt attached with a surcingle or girth around the horse's barrel for increased security and comfort. Xenophon mentioned the use of a padded cloth on cavalry mounts as early as the 4th century BC. The saddle with a solid framework, or "tree", provided a bearing surface to protect the horse from the weight of the rider, but was not widespread until the 2nd century AD. However, it made a critical difference, as horses could carry more weight when distributed across a solid saddle tree. A solid tree,
Girth (graph theory) repeating the Mycielskian construction used to form the Grötzsch graph produces triangle-free graphs of arbitrarily large chromatic number. Paul Erdős was the first to prove the general result, using the probabilistic method. More precisely, he showed that a random graph on vertices, formed by choosing independently whether to include each edge with probability has, with probability tending to 1 as goes to infinity, at most cycles of length or less, but has no independent set of size Therefore, removing one vertex from each short cycle leaves a smaller graph with girth greater than in which each color class of a