More intoxicated; having consumed more alcohol than someone else or than before.
From 'drunk,' the past participle of Old English 'drincan' (to drink). 'Drunker' is the comparative form using the '-er' suffix to compare degrees of intoxication.
English speakers actually argue about whether 'drunker' or 'more drunk' is correct—the comparative forms of adjectives in English are inconsistent ('faster' vs. 'more beautiful'), and this reflects the messy history of English absorbing words from multiple language families.
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