This work derives an upper bound on the maximum cardinality of a family of graphs on a fixed number of vertices, in which the intersection of every two graphs in that family contains a subgraph that is isomorphic to a specified graph H. Such families are referred to as H-intersecting graph families. The bound is derived using the combinatorial version of Shearer's lemma, and it forms a nontrivial extension of the bound derived by Chung, Graham, Frankl, and Shearer (1986), where H is specialized to a triangle. The derived bound is expressed in terms of the chromatic number of H, while a relaxed version, formulated using the Lov\'{a}sz $\vartheta$-function of the complement of H, offers reduced computational complexity. Additionally, a probabilistic version of Shearer's lemma, combined with properties of the Shannon entropy, are employed to establish bounds related to the enumeration of graph homomorphisms, providing further insights into the interplay between combinatorial structures and information-theoretic principles.

翻译：本研究推导了在固定顶点数上，一族图中任意两个图的交集均包含一个与指定图H同构的子图时，该图族最大基数的上界。此类图族被称为H-相交图族。该上界是利用Shearer引理的组合版本推导得出的，构成了Chung、Graham、Frankl和Shearer（1986）所得上界的一个非平凡推广，其中H被特化为三角形。推导出的上界以H的色数表示，而利用H补图的Lovász ϑ函数表述的松弛版本则提供了更低的计算复杂度。此外，结合香农熵的性质，应用Shearer引理的概率版本建立了与图同态枚举相关的界，从而进一步揭示了组合结构与信息论原理之间的相互作用。