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An $n$-vertex graph containing no path of length $t$ has at most $(t-1)n/2$ edges.

This path-length bound is a different result from the Erdős-Gallai theorem on degree sequences.

The Erdős-Gallai theorem characterizes which finite nonincreasing sequences of nonnegative integers form the degree sequence of a simple graph, using an even total degree and partial-sum inequalities.