{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:45Z","timestamp":1753893825136,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Let $n$ be a positive integer, $c$ a real positive constant, and  $p(n) = c\/n$.  Let $U_{n,p}$ be the random unary predicate under the linear order, and $S_c$ the almost sure theory of  $U_{n,{c\\over n}}$. We show that for every first-order sentence $\\phi$: $$ \tf_{\\phi}(c) = \\lim_{n\\rightarrow\\infty}{\\Pr}[U_{n,{c\\over n}} { has\\ property\\ } \\phi] $$ is an infinitely differentiable function.   Further, let $S = \\bigcap_c S_c$ be the set of all sentences that are true in every almost sure theory.  Then, for every $c&gt;0$, $S_c = S$.<\/jats:p>","DOI":"10.37236\/1308","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T20:53:26Z","timestamp":1578689606000},"source":"Crossref","is-referenced-by-count":1,"title":["Limit Probabilities for Random Sparse Bit Strings"],"prefix":"10.37236","volume":"4","author":[{"given":"Katherine","family":"St. John","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[1997,10,14]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v4i1r23\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v4i1r23\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T01:11:32Z","timestamp":1579309892000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v4i1r23"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1997,10,14]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[1997,1,1]]}},"URL":"https:\/\/doi.org\/10.37236\/1308","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[1997,10,14]]},"article-number":"R23"}}