The s-flag-transitivity of designs

The s-flag-transitivity of a design is an extension of the flag-transitivity. In this paper, we first introduce the basic concepts and properties of s-flag-transitive designs in detail. Then, when the s-flag-transitivity of the design can be extended to any positive integer s' less than s, this 性质 is referred to as the downward hereditary with respect to s, and the corresponding design is abbreviated as a hereditary s-flag-transitive design. We prove that the upper bound for the parameter s of hereditary s-flag-transitive designs is 5. Finally, we completely classify the hereditary 4-flag transmission designs and 5-flag transmission designs.